Throughout the Principate and Late Antiquity, the emperor was the single greatest landowner in the Roman world. Collectively referred to by scholars as the imperial estates, the emperor directly owned vast tracts of farmland scattered across the Empire, as well as mines and forests. Imperial estates began as the private property of Augustus, but grew substantially over the centuries through conquest, confiscation, and the adding of additional private holdings by each dynasty, as each new imperial family brought with them (often substantial) landholdings. Many Roman peasants thus would have been tenants of the emperor himself, living on and working his personal holdings. As landowning offered the most reliable and consistent revenue stream in the Roman world, emperors had access to significant private funds. While most of the revenue from the imperial estates was funneled directly into the treasury (a process overseen in Late Antiquity by the bureaucratic department of the comes rerum privatarum), the privy purse, and the expenses of the imperial household, came directly these personal holdings. It is worth noting that only a fraction of the revenue required by the Roman state came from the imperial states. The vast majority came through varying forms of taxation. So yes, Roman emperors were quite personally wealthy, although reliant upon taxation to fund the army and state apparatus.

/u/HatMaster12 already gave a great answer to this, but there are a few things I would like to add.

The "treasury" was not a separate entity from the emperor's own wealth. State funds were the emperor's personal funds.

So to answer your question, no they did not receive a salary. Their own life style was a state expense, everything they or their families spent money on was a state expense. Taxes were essentially determined by how much the emperor needed in money outside of their imperial estates.