I have been in this situation a few times.
If company (A) sends you to company (B), the following things may occur:
- You may receive a cheque directly from Company A
- Company B may not know how well you are compensated
- Only very rarely, your placement at that company may be confidential
Unless your placement is secret, you can totally list on your resume, or your LinkedIn, that you worked at whichever of the companies you wish. You can even list both, since you did technically work for both.
For example, Company A, RecruitoCo, hires you, and sends you immediately to work at respected developer DevTech.
You get a cheque from RecruitCo, but on a day-to-day basis you work for people at DevTech. So who do you say you work for? Barring any contractual provisions, you can identify as working for either. Whatever suits you best.
Legally, you are bound by any covenants of secrecy that are within your contract, so I'd advise you to look at it, if any particular detail concerns you.
Source: Worked this way since 2013.