Learn Russian: Мы потеряли деньги, зато получили опыт. - We lost money, but on the other hand we got experience.

While но just means a simple 'but', зато carries the specific meaning of compensation or a silver lining. It translates best to 'but on the other hand' or 'but to make up for it'. In this sentence, the good thing (gaining experience) makes up for the bad thing (losing money).

Is it mandatory to drop мы before получили?

It is not strictly mandatory, but it sounds much more natural to drop it. Just like in English, you do not need to repeat the subject if it has not changed between the two actions.

Both потеряли and получили are in the perfective aspect. Why?

Perfective verbs are used for completed actions that produced a result. The speakers are not focusing on the ongoing process of losing or receiving, but on the final outcome: the money is gone, and the experience has been acquired.

Why do the noun endings for деньги and опыт look like their dictionary forms, since they are the direct objects?

They actually are in the accusative case! For inanimate masculine nouns like опыт (experience), the accusative form is identical to the nominative (dictionary) form. The same rule applies to inanimate plural nouns like деньги (money).