Many universities have visiting (also sometimes termed adjunct outside the US) faculty positions for non-salaried academics associated with a department. My question concerns the cases where faculty members, rather than an adjunct appointment, hold parallel appointments at two different universities, perhaps as a part-time appointment. What types of rules are there regarding e.g. attendance, affiliation in publications, teaching, PhD supervision, committee membership, access to funding, lecturing responsibilities, and salary for such cases? I'd appreciate pointers to corresponding policies or regulations.
What types of rules are there regarding e.g. attendance,
Contractual: Attendance requirements are defined by employment contracts.
affiliation in publications,
List both, since you are employed by both. Perhaps order affiliations by who paid for the work, or include an acknowledgement to the institute that funded the research.
Your institutes will likely have expectations, either explicit or implicit. You'll have your own expectations.
Likely institutional expectations, e.g., you must serve on at least one committee.
access to funding,
Apply using the most favourable institution
I don't understand this point.
You can negotiate with each institute.
I'd appreciate pointers to corresponding policies or regulations.
I think in practical terms this is mostly decided by the first employer. So you already have an appointment at one university and think about starting a second one at another university. Then you need to check your work contract with the first university and ask them what kind of side job they permit. That is the relevant authority and in general whatever they permit is fine (you can and should negotiate with them).
Apart from that you also have to obey general labour laws, so for example in most european countries you couldn't have 2 full time contracts because total weekly work time is limited. AFAIK the US doesn't have this restriction but there might be others that are relevant.