What is the difference in performance between a two single-field indexes and one compound index?
I'll answer on the MongoDB side (no Postgres knowledge, sorry). The classic answer is that a query only ever uses a single index, so if you have a compound index, good data cardinality and are not using multiple $or clauses, then the compound index will significantly outperform the two single indexes. Some exceptions to this would be those $or clauses (each clause uses a single index and merges the results, so for small data sets this might be a wash) and a data set where a single index fits in memory but the compound index does not.

MongoDB supports intersection as of version 2.6 but the only reliable way to tell if it will select intersection as a viable strategy is to have a sample data set and indexes and try it out. Compound indexes are a far more reliable strategy for now, and would be my recommendation for searching on two fields generally with MongoDB.