Should a bank run an OFAC check for both incoming and outgoing wires? Also, should the receiving/sending bank be checked (e.g. if one of your customers sends a wire to another individual whose account is with Chase do you OFAC the two individuals only or do you need to check Chase as well)?

In my opinion, yes, incoming and outgoing, all parties, all banks, all obi and bbi notes.

The decision of who to run against the OFAC list is a matter of your bank's risk assessment. I believe that most, if not all, banks run every name in a wire record (individual, entity, bank) against the OFAC list to be on the safe side.