Homeschool critics, people considering homeschooling, concerned grandparents, and the simply curious tend to ask the same sorts of questions about homeschooling — and not, usually, positive ones that acknowledge the many benefits of homeschooling. The most popular questions are ones that ultimately amount to some variation of what about socialization?
The term's meaning is twofold, referring both — as the dictionary tells us — to "the process of learning to behave in a way that is acceptable to society" and "the activity of mixing socially with others" . These two things are of crucial importance to a person's healthy development. We humans are deeply social and co-operative beings, after all, and those wondering how to meet a homeschooled child's social needs are probing a valid subject.
Research About The Socialization Of Homeschooled Children — And Its Flaws
The answer is a resounding "yes". Homeschooled children can have their social and academic and intellectual needs met, thrive, and grow up to become productive, sane, and content adults. You don't have to take my word for it; there's science to back this particular claim up, and to prove that to you, I'm now going to hit you over the head with some fairly mindlessly selected studies. It might sound like propaganda to you, in part because some of it is, but bear with me.
A 2013 review of studies researching the socialization of homeschoolers found that homeschool parents raise their children to respectfully interact with a diverse group of people, prioritize social interactions outside the family, and believe their kids' social skills are on par with public school kids. Homeschooled children, the review suggests, tend to have better-quality friendships and more harmonious relationships with parents and other adults than their publicly schooled peers, as well as possessing a strong sense of social responsibility. Homeschooled children are generally optimistic and happy people, the paper concluded. 
"Arranging opportunities for their children to socialize with others" poses one of the greatest and most frequent challenges among unschoolers, a subset of homeschoolers, another study found. It is a difficulty they apparently overcome, because the paper concluded that "improved psychological and social wellbeing for the children; and increased closeness, harmony, and freedom for the whole family" represented the most significant benefits of this educational method among those analyzed. 
A fascinating study that explored the impact homeschooling may have on the concept of good citizenship noted that citizenship education in schools mostly comprises "history, geography and social studies lessons, with some limited participation in extra-curricular activities both inside and outside the school". Among homeschoolers, meanwhile, "participation in the public sphere is a more important component of their education". They "are much more involved in things like volunteer work than schooled children" — we're talking 30 percent vs six to 12 percent! — "which also further offsets socialization criticisms". In the process, homeschoolers engage in "a process of defining their own vision of what it means to be a citizen". 
- "Basic demographic data are unavailable. Every state in the U.S. has its own unique homeschooling law, and states approach data collection in a very haphazard fashion."
- "...much of it is politically motivated, particularly in the United States context. A large number of studies, especially those most frequently cited in popular accounts and in the media, have been performed under the auspices of a prominent homeschooling advocacy organization: HSLDA..."
- "Many university-housed academics who have published on homeschooling have come out clearly as critics of or advocates for homeschooling." 
Or in layman's terms, many homeschool studies simply suck. Because unbiased data is unavailable. Because study participants are often self-selected and even if they weren't, you'd never catch all homeschoolers since many don't prefer to be scrutinized by those who may use whatever conclusions they make against them. Because study authors have their own agendas and set out to prove a predetermined outcome — which, folks, is not how science actually works.
If I wanted to make religious homeschoolers look bad, I could have quoted a study that found religious homeschool dads think it's their duty to "nurture the home as a 'protective cocoon' where concerted moral cultivation can take place", in order to create a "system of 'total socialization' aimed at negating the influence of competing socialization agents" . In other words, to firmly lock the windows and doors to prevent secular winds from corrupting their kids and creating a mini totalitarian regime in their homes?
If I wanted to make secular homeschoolers look bad, I could have quoted a study that found "homeschoolers with weaker religious ties were three times more likely to report being behind their expected grade level", "two and a half times more likely to report no extracurricular activities in the prior year than their traditionally schooled counterparts", and "more likely to report lax parental attitudes toward substance use". Its conclusion? "Findings suggest homeschoolers with weaker religious ties represent an at-risk group."  Lax attitudes toward substance abuse probably include modeling that it's OK to drink a few beers on Friday night?
Why The Answer To The Socialization Question Lies Within Our Own Families
Why didn't I spend more time selecting the best possible studies and presenting them in a more nuanced way? Because the research sufficiently demonstrated that homeschoolers can have a rich social life and be socialized, but that doesn't mean anything particular for any individual. The fact that some people (Turpins, anyone?) use homeschooling as a cover for all sorts of abuse, including social neglect and much worse, doesn't reflect on those who have large and diverse social networks. The fact that some homeschool kids have a whole bunch of quality friendships doesn't make things any better for that depressed teen in a small village who really doesn't get along with anyone.
We, homeschoolers, are a diverse bunch.
We're LDS with 10 children, and Pagan, lesbian, tattooed, vegans homeschooling onlies. We're born-again Christians convinced that Harry Potter and sex ed are tickets to hell, and atheist radical feminists on a war path to smash the patriarchy. We're expats who know our kids would never fit into a local school, and families who simply know childhood is fleeting and want to be as present as possible.
We're raising children with autism, with dyslexia, with Down Syndrome, gifted children, kids who are so busy with their professional ballet career that they simply have no time to fit "regular school" into their schedule, and completely neurotypical children. We're urban and rural, introverts and extroverts, philosophically opposed to public school and staunch supporters of it even if we chose differently ourselves. We started researching educational philosophies long before our kids were born already knowing we'd homeschool them, and we just pulled our child out of school due to severe bullying last week, having literally no clue what we're doing.
We follow all sorts of educational paths, from unschooling to classical education. We belong to homeschool co-ops that actually oversee most of our kids' schooling, our kids are dual-enrolled in community colleges, and we learn and home and beyond without paying third parties. We're offering a rigorous education that would cause some to accuse us of tiger parenting, we're frankly so lax that some within "our own ranks" would accuse us of educational neglect (because we think helping with infant care or playing Minecraft all day, every day, counts as learning), and anything in between.
Yes, I've actually encountered every single one of these people.
Homeschooled children can indeed grow up to be happy, optimistic, socially well-adjusted, and engaged in their communities. As parents, we can encourage, but never guarantee, this outcome by doing things like actively and broadly educating them, helping them seek out social opportunities (and when they get to the teen stage, perhaps just not getting in their way) and finding a place within their communities, and being loving and relatively sane ourselves.
It might take tremendous work to provide your children with social opportunities, or they may present themselves. Your child might be a normative social butterfly or a quirky aspie or gifty who gets along better with adults than children. While it is true that brick and mortar schools provide close proximity to same-age peers by default, and that close proximity may lead to a fulfilled social life, any child who doesn't fit in, sometimes to the point of bullying, knows it can be a double-edged sword. Homeschooling doesn't ultimately say that much about a child's social and other development at all — a combination of what you and your family make of it and your child's own personality do.