Q: Who found whom?
A: They both found each other - Jesus found a disciple and Philip found a Messiah.
Overall, this question is essentially based on a translation issue - there is a tacit assumption that the meaning and grammar associated with the original word translated as "found" is essentially identical to how we understand this word in the context of the english language. While understandable, an assumption like this is not particularly helpful in general, and particularly so in this case. The actual word in question is:
Greek transliteration: heuriskō
(act.) to find, discover, meet; (mid.) to obtain; (pass.) to be found
to find, to meet with; to find out, to detect, discover; to acquire, obtain, win, gain; to find mentally, to comprehend, recognize; to find by experience, observe, gather; to devise as feasible (source)
In english, 'found' is nearly always used in a transitive sense with a well defined distinction between subject and object, eg: "I [subject] found it [object]". One of the alternative translation choices - 'meet' - is capable of being used intransitively however, eg: "we [subject/object - ie ill-defined object] met yesterday", and if it is used in a transitive construction grammatically, the semantics don't necessarily demand strict choices for subject and object - grammatically, there is a difference between "Harry met Sally" and "Sally met Harry", but as far as semantic distinctions go, you'd be scraping the bottom of the barrel to find one.
If you're a whiz with ancient/koine greek grammar, perhaps you could make more of this issue than I can, but to my mind, we should be wary of unnecessarily projecting a transitive sense into the grammar. Additionally, the alternate translation options, imply that even if the underlying grammar implies transitivity, there may be no particular semantic significance to that transitivity. In conclusion, don't read so much into it.
The short answer I've given above is based on understanding the overall context of the passage - it's relatively agnostic regarding the 'correct' translation choice of εὑρίσκω (just following the crowd) and how strictly transitive the grammar is, it merely relies on the fact that both εὑρίσκω and 'found' can be used with varying levels of significance.
Re the 'we' issue, a somewhat unlikely possibility is that Philip was the one unnamed disciple of the pair mentioned in vv35-40, and his 'we' just meant himself and Andrew. A more likely option, is that Andrew, the unnamed disciple, Peter, Philip and Nathaniel knew each other well enough to think of themselves as a group, and that by that stage, Philip had found out about the other interactions of the group members with Jesus, while Nathaniel had been off doing something else. According to this interpretation, 'we' refers to the rest of the group apart from Nathaniel.