All truly large cities in the ancient world needed an outside grain supply. (And therefore access to the sea.) That's simple logistics: you can't feed a city of a million inhabitants (like Rome, according to our best guesses for the 1st century AD) with just local produce.

However, the important thing to remember about the Roman Empire is that cities of that order of magnitude were extremely, extremely rare.

Specifically, there were 4 to 5 cities with hundreds of thousands of inhabitants: Rome, Alexandria, Antioch, Carthage, and in the late empire Constantinople. Alexandria and Constantinople had up to 500,000 inhabitants each at their heights, Carthage and Antioch probably had as much combined according to Goldsworthy's "fall of the West." (Which isn't the best source on this topic, but the one I have handy right now.)

Carthage and Alexandria were situated right inside some of the most productive agricultural regions. Constantinople was reliant on grain shipments in the same way Rome was (and also had a grain dole.) I'm not actually sure where Antioch got its food. For some reason the Oxford Dictionary of the Classical World doesn't even have an entry on the city. (It's abbreviated from the full OCD, sure, but you'd think Antioch would warrant inclusion...)

Most other cities in the empire only had tens of thousands of inhabitants, with perhaps a handful more approaching the 100,000. Smaller cities like that could subsist on grain produced in their more immediate hinterlands far more easily.

Edit: It's also important to remember that even if you put all these settlements together, you only have a minority of the total population. The vast majority lived in the countryside. So most of the grain produced in the Roman Empire was actually eaten by the people who produced it. There was only so much surplus.

And as for agricultural grain-producing provinces in general: the big ones were Sicily, Africa, Egypt and to a lesser extent Sardinia and Corsica. Even in imperial times, the majority of grain consumed by Rome came from Sicily and Sardinia and Italy itself, which was then supplemented by the produce from Africa and Egypt for the grain dole, according to the OCD.