In general, pluralism of beliefs was absolutely tolerated in the Roman Empire, and its only really when Christianity became really influential that people had to start justifying their beliefs at all. As to whether the Romans persecuted Christians, well, that's another question...

It's incredibly clear from the ancient sources that different people believe different things in the Roman Empire. It's important to remember that Roman religions didn't have a book that explained what to believe (an orthodoxy) and nor did the component cults. There were certainly lively philosophical and literary debates on religious beliefs, ethics and cult practices, but there is generally not the same 'I'm right and you're wrong' vibe as you get from religions based on a central text.

Take one example: Cicero, in general verrrry happy to be confident that he is more correct than anyone else, talks about the worship of images in his De Natura Deorum (1.77-83). Here he goes through the different beliefs of Romans at different stages of history, the possible theories behind the practises (maybe people think they can access the gods through images, maybe because of human vanity, maybe the limitations of human skill...), he also describes Egyptian beliefs and their portrayal of animals as divine in particular. But he never says here "and the right answer is xxx", or even "and they are wrong". Other eminent writers on the same issue can take wildly opposing views (Pausanias will in general talk deeply reverently and admiringly about religious images, Plutarch will deeply criticise them) and within the context of the religions of the Roman world, nobody is being blasphemous and nobody is offending anyone else's view of the divine.

As Charles King would say:

"The Roman Pagans did not merely lack the Christian focus on orthodox sets of beliefs, but possessed specific alternative mechanisms for the organization of beliefs that allowed clusters of variant beliefs to exist within Roman society without conflict.”

This is one of the reasons why its difficult to pin down what general religious beliefs might have been in the height of the Roman Empire - there's no need for a consensus in that context. There's a wealth of literature on different ancient authors' religious views, and I'd strongly recommend Davies's 2004 book Rome's Religious History for a rigorous analysis of some of the most important historical sources for Roman religious practices.

There's also the scope of the Empire to consider - the Roman world incorporated many varying belief systems that simply had to coexist in one way or another. The Greek and Roman religions, for example, were so effectively made to be comprehensible to one another that they are now often thought of as the same thing, although to their believers they were pretty fundamentally different in many ways. Take on Near Eastern, African, Celtic and more religious traditions and you've got a system where hybrids and overlapping beliefs are the norm, rather than the exception.

This could extend to 'importing' foreign gods. There was a tradition of 'evocatio', in which invading generals could ask the local gods for their support (i.e. asking them to abandon their city) and after victories it was common for the items of the cult, and sometimes the worship of it, to be transported to Rome (see Pausanias, 8.46.1 for Augustus's massive movement of cults/cult objects to Rome; Livy 26.34.12 (Images and statues given to a college of priests after the siege of Capua to decide which were sacred and which profane); Polybius, 21.30.9 (M. Fulvius Nobilior, having taken Ambracia released the Aetolians under their rule and carried off their cult images, statues, and paintings). We also have plenty of physical evidence for imported cults - the materials found in Pompeii for the cult of Isis, for example, make it clear it was a popular cult there.

I could go on... but hopefully it's clear that there was variety and flexibility in the religions of ancient Rome in general.

As for the persecution of Christianity...that has a big and complicated historical tradition associated with it. Certainly, Pliny's Letter to Trajan (Epist. 10.96) makes it clear that Christians were executed. But was it for being Christian or was it for rejecting the worship of the Emperor? Suetonius and Tacitus may also make reference to punishment and persecution of Christians, but again its not clear how much this has to do with disagreeing with their beliefs as such. The complex historiographic tradition has a lot to do with the personal religious beliefs of authors here, and it's delicate ground. I'd recommend Moss's The Myth of Persecution and de Ste Croix's Christian Persecution for a good overview of all the passionate arguments.