I think it's fairly safe to say that a highly educated person from (insert country name) has a pretty good general understanding about their country's history, or at a minimum, could look most of it up. How would an educated Roman compare? Would their knowledge be of facts, or would they "know" that Rome was founded by Romulus and Remus, sons of Mars?
First off, it depends when you are referring. Secondly, your use of the phrase "facts" implies to an extent that we know now the truth, which we regrettably do not. A large reason that we do not is because Myth and legend were as real to later Romans (say, late republic onward) as true history, just as religion was prominent in daily life, whereas in the modern western world today both myth and religion are less entwined in our perceptions of ourselves.
So the idea of a foundation myth such as Romulus and Remus in 753 BC, the sons of a god who were suckled by a wolf etc would sound propesterious to (most) of us, and indeed the archeolgical evidence shows that this is not the case. But it is safe to say that anyone who lived in Rome would know this story and believe it, for what reason is their to doubt it? Romans too would know of the period of the Kings (although this is certainly simplified and a condensced version of the reality) and the period of the early republic. I tend to think that until you get historians writing about the Romans (intially the Greeks) you have to take everything that happened as a flawed truth. Things are altered, and changed from the truth, but not drastically because we can see the effects of certain events later on. For example we know that the Gauls did sack Rome, but the later Romans who write about it (Livy) are recording it centuries after the fact, after it has been romanticized and altered.
It is worth saying that the Romans had a very nostalgic view of the past. The later Romans were always looking back to a golden age where Romans were true Romans who obeyed the law and honoured the gods and their family, and put the state infront of their own ambitions. The senators especially later on, longed for a return to the past. Cicero seems to almost wallow in a desire to return to the past.
But essentially: Although we can not know the extent of a Romans knowledge of the history/myth of the empire at any point in time. They would have thought that they had (if they were educated) a very good idea of what was to them, Romes History.