It depends on the city. In busier areas, the radio report for a routine, stable patient just takes up time. Their volume is so high already, an extra patient isn't going to make or break anything.
In my area, we are required to make an entry report with Pt age/gender, CC, brief HOPI, VS, any treatments, and an ETA.
If hospitals get busy and we go to "The Plan" then no, dispatch lets the hospital know when they have a unit coming in, and if they have a special need (STEMI alert, Stroke alert, Burn, Trauma and Level, etc). If the hospital wants to contact us, they have access to "The Plan" Transport channel which all units must monitor when we are on "The Plan."
An aside: "The Plan" sounds very ominous and indeed it is--we have only had one instance that we've been on it in the past 5 years, but it doesn't sound as ominous as "The Other Plan" colloquially known as "Plan B" though the county didn't want to call it that because it sounded like things are really bad (which they would be if we were on Plan B).