It is true yes.
If it is a perm role with a company that you are being introduced to by the agency, it should be the company that employs you and therefore your contract is with the company.
Of course, if you are being employed in a job with the agency itself, then your contract is still with them but is still a permanent one.
The finders fee paid by a company to an agency is nothing to do with you; the agency will have established an agreement with the company where they supply a person and in payment for that they get �X amount (usually a percentage of your salary) and then everything to do with your employment (with the possible exception of negotiating salary) is done between you and the company.
There may be a clause in the agreement that if you leave the job within a certaim amount of time, the finders fee (or a percentage of) is repaid by the agency because of the fact that you're no longer in the job but again, that's nothing to do with you.