In short, yes they generally did. Unlike us the Romans didn't really divide the social sphere into either secular or ecclesial affairs. This concept didn't start to emerge until Medieval Popes and kings started sparing over things like sovereignty and jurisdiction. In fact, the Romans genuinely believed the gods were part of the political process and, given that the Gods had particular rights, they had a very public presence Roman social life. In particular, the Gods had property rights that were either greater (locus sacer) or equated the property rights of the average Roman citizen. Extending upon this, Romans often lend space to the Gods in their own household, often in the form of altars, murals, and sculptures. In essence, the Romans believed that the gods quite literally had a physical presence that had to be respected.

Source: Jorg Rupke, "Communicating with the Gods" in A Companion to the Roman Republic, ed Nathan Rosenstein and Robert Morstein-Marx (Blackwell, 2010), 215-235