Who determines the details of the offer, the salary etc.?
The answer does varies organization to organization. Generally it's the hiring manager (the person you report to or eventually report to) that you are negotiating with for your salary, signing bonus, relocation, title, etc, but the manager may be constrained by their budget and other company policies such as pay bands for certain levels and titles that are typically set by HR.
You may end up negotiating with someone who's more senior than your manager. I interviewed in a team where all salary negotiation went through the senior directory of engineering even though I ended up reporting to the engineering manager (senior director of engineering > director of engineering > engineering manager).
A recruiter typically serves as a buffer between you and the hiring manager helping coordinate details of your interview and may sometimes negotiate salary to a certain degree. If you want more than the recruiter is authorized to give, you would have to negotiate with the hiring manager. If you want more money than the HR guidelines, a manager may have to go through and request an exception or request a change in your title. I once requested a government organization match a private company's salary offer. While the hiring manager agreed to try to match, the salary match was eventually rejected by HR, because it violated the pay band for my level.
Does the team also play a part?
No. The team typically only will advise hire or no hire and sometimes what level is appropriate (entry-level, senior, staff, etc) for the candidate. The team does not know what the individual's salary nor do they participate in the salary negotiation.