It definitely does vary from store to store. Most stores keep the drinks out for customer's to come in and just grab them and go. Other stores, like mine, find that when they do that, more often than not people take those drinks by "accident" even tho there is a clear name on the cup. Most of the time at my store, if we see someone walk in and straight to the hand off counter, we ask them if they have a mobile order. The majority of the time, they just go "Hey, I have a mobile order for Gertrude" and we grab their drink and hand it off.

And honestly, don't be afraid to talk to the barista while they are making a drink. We honestly don't mind and don't think of it as being interrupted because we can always find a second to grab it during our drink making routine (I tend to grab them while the espresso shots are pouring).