The JOSE standard introduces a signature (JWS) and encrypted data (JWE) which describe a data bundle, either signed or encrypted. However I cannot find a "request of signature" or "request of encryption" (it's possible to use JWE as request for decryption).
I would like to ask a remote end, as a proof of shared secret ownership, "sign me these bytes using this algorithm and this key" for which the response is a JWS as well as "encrypt these bytes using this key" for which the response is a JWE.
I'm trying to not reinvent the wheel here and while reading the JOSE spec I couldn't find anything like that that could be of use. It seems like such a common type of request that I wonder why/how it could be left out.
You're correct, there is nothing in the standards for this type of thing.
I'd question why you want to do things this way. Is there something significant in the content of the payload you want them to process? Why would it be unsuitable for the remote end to just send you a JWT?