I was excited to hear that "my student"* Justin Thaler, working with Salil Vadhan and his student Jon Ullman, had a paper accepted to ICALP (Faster Algorithms for Privately Releasing Marginals). It's a great example of something that's been going very nicely here at Harvard: students (+advisors) crossing boundaries and collaborating together on problems.
I know it's a common refrain that people don't realize that computer science research is, generally speaking, a highly collaborative enterprise, but it's certainly worth repeating. Most projects aren't done by one person coding or mathematically proving alone, but working as part of a team. Graduate students should know that and embrace it. There are huge advantages of this approach. The natural one to point out is the great synergies in developing and working through ideas from multiple people with different skill sets and points of view. You also learn about new things that can turn out to be useful to you in unpredictable ways by working on projects outside your standard box. But perhaps less often mentioned is that working with others is generally just much more fun, and I think it's easier to produce good and great research when you're having fun doing it.
I feel like that's stating the obvious, but thinking back to my time in graduate school, maybe it's not. Collaboration in the theory group certainly wasn't frowned upon, but it wasn't entirely actively promoted, either. Micah Adler and John Byers worked very successfully together on several papers early in their graduate student careers, and I remember working on some problems with multiple friendly older students. But I also recall, while not what I would call a competitive atmosphere, the sense that you really had to prove yourself "on your own", especially in putting together a thesis, leading to at least some cases where "credit" became a issue to some people. The theory group was pretty social, so overall the joys of working together probably won out overall. To me today feels like a non-trivial delta from that (here at Harvard at least, but more widely as well), though perhaps to the students aiming to graduate and get jobs it doesn't feel that way -- I can imagine, depending on the setting, that the pressure to do something great on your own is still primary.
Maybe the most encouraging thing I can try to say, which really clicked solidly in my head this recruiting season, is that one thing I look for in a candidate is whether they can work well with other people -- not just their advisor. I think that's important, both for the candidate's future success and for CS at Harvard. I don't think there's a formula I have in mind to weight "individual success" vs. "group success", but I'd be more skeptical of a candidate that didn't have signs of both. Further, I recognize that the talent of the individual can shine through on group projects -- while the ability of a candidate to work within groups can't really shine through on individual projects.
*Increasingly a misnomer, as Justin's well beyond being "my student" in any meaningful sense at this point, but it's still the most convenient reference.