I'd say that anything, from having a social life to learning calculus, is really up to you to accomplish, especially to accomplish well. Beyond the elementary level different school systems are mostly just different support networks for all that; and the homeschool version depends greatly on what sort of homeschool community is in the area and whether you're willing and able to hook up with it. (I've known some so massive and active that they rivalled the public school communities in the area and even each other in a friendly, many people participate in both sort of way.) Granted public school has both more drudgework -- and more things that you would always have to jump through hoops to get but at least the hoops are positioned facing the official institutions so they don't tell you, "Sorry, you can't jump through the hoop," without even letting you try.
But there's a funny little fact: if you don't pursue some high-end or cutting-edge career, chances are you'll retain what you need to from school, get rusty on the rest and learn anything you missed on the job. And here's another one: if you do pursue a high-end or cutting-edge career then chances are good that, while you might get in cheaper by getting scholarships right out of high school, most of the skills and learning you need to acquire begin (or at least begin to be truly fortified or applied) at college anyway. In the end, it's possible that the thing most affected by whether you do public school or homeschool for a year is -- where you happen to be that year.
Personally, I don't have a lot of the opportunities people said I could pursue as I came up out of high school and into college. But I have independence now. And independence means that I work a full week and have little time for my own pursuits, but I can spend that little time pursuing -- whatever the heck I want. Do you know how that compares to the big opportunities everyone used to talk about? It's AWESOME. Oh, it may not get me any recognition, but accomplishment's another matter. And it's entirely on my own time and terms, aside from the mandatory full work week. And hey, that full work week? Everyone who isn't the spawn of a millionaire has that anyway, some people just manage to get to do what they actually like for it -- and I managed to pull that off too, despite switching colleges so many times it took me four years to get a two-year degree.
So I say, encourage your friend to do whatever he wants to do, just so long as he makes the most of it! And as others said, don't be afraid to try to do some of it with him, whether for your own betterment or just to keep up.
(Full disclosure: I was homeschooled from the start up through college. I think it went well; I got to college and felt like many [not necessarily most, but many] undergraduate classes were just remedial highschool, and I graduated and am still in the area I grew up in and know a lot of people here -- both of my own generation and of the older -- because we have been part of the same homeschool community. But most importantly, I got to that point where I can continue growing and teaching myself, meeting new people and getting to know them too, and basically able to make up anything lacking in my background and experience through further effort; and once you're there I don't think it matters much where you got there from.)