Tobias- The question is a general one. One needs to know the specific infections then only one may be able to determine whether the immunological responses are interactive or separate.-- Kanti
In principle, one infection, or the other, or both stimulate the host mechanisms for the immunological response. In fact, generally, they boost the host response due to the activity of the response system that is already triggered by one, helping the other. Research is limited, does generally not mean that the already searched, or already present literature is the truth. Everything is changing, including the principles of host and infection relationships over a period of time.
It depends on the type of infection and what type of innate immune cells are activated. TLRs and other innate receptors present on immune cells play an important role in determining the response to an infection. So a co-infection may be synergistic to first infection in terms of inflammation but it's too complicated to understand theoretically. It depends upon also which organ is involved, e.g lung vs GIT vs systemic. Immunology is such a vast and changing research area where new concepts are emerging almost everyday, so it's difficult to say what's likely to happen until you test in lab your hypothesis.