Strictly, a 200 only means that the request was successfully processed. This is regardless of the actual outcome of the call from a business perspective.
You are in effect relying on a convention of "we will throw an exception or otherwise fail the call if the user is not authenticated", to authenticate users.
Depending on the convention, you could conceivably have a scenario where a user was unauthenticated but the call was still successfully processed.
From this perspective, it may be worth having the service2 return a response which could then be interrogated to close this circle.
Alternatively, you could have the client call the authentication service directly, retrieve a token, and then present this token with any other request. This would mean that service1 is no longer responsible for having to know that a caller is authenticated.
The question is whether or not to then test in Service 1 each time the
response is received
Sorry, I would appear to have misunderstood the spirit of the question somewhat.
I am slightly confused - are you asking that, if the system under test is service1 then should any response from service2 also be part of that test?
I would say you would have to have some test which could prove that the sercvice2 response interrogation logic is correct, but this could be done at the unit test level. I don't think you'd need to do this for tests running against a deployed instance of the service, which is by nature more about the service behaviour at the boundary rather than internally.