Do contractors need to sign any contract with Remote?

Understanding whether Remote is directly involved in your contractor agreements depends on which service you're using. Here's what you need to know:

Contractor Management:

No. The contract is between your company and the contractor. Remote is not a party to that agreement⁠.

Contractor of Record (COR):

Yes. The contractor signs a Contractor Services Agreement (CSA) with Remote or a Remote-designated subcontractor as part of the COR engagement⁠. 
See also: What is Remote Contractor of Record?