Does the elephant get bigger when I bring in the ballista?
Yes, you get the counter on Trumpeter.

When an ability is looking out for counters being placed on an object, the game also counts cards that enter the battlefield with counters as having had those "placed" on it: it will trigger abilities like Trumpeter's, Hardened Scales would give the Ballista another +1/+1 counter, etc.


  121.6. Some spells and abilities refer to counters being “placed” on an object. This refers to putting counters on that object while it’s on the battlefield and also to an object that’s given counters as it enters the battlefield.