When xp had (court ordered) contact, she was supposed to come and pick up DD at our house, and return her.
I would find myself getting quite worked up waiting for her to arrive (invariably late), and of course this rubbed off on DD - 3 to 4 at the time. It turned out that some / most of DD's anxiety was actually down to crap parenting by her mother, but that's a whole other thing.
In due course she asked me to pick up and drop off DD at her flat, and that was easier as at least I could drop off DD at the right time and be there to pick her up at the right time.
If I could've relied on xp to consistently be at a given place at a given time to have contact, then I would've preferred to do hand overs at a coffee shop, as that would've been a bit nicer all round - comfy, neutral, you wouldn't mind (too much) if they were running late, far less chance of a scene being made (including by DC), etc etc.
Point is, it depends on how civil things are between you and your XP, and how consistent and reliable he is with regard to your DD. It also depends on how convenient it is for you (and him) to meet at Starbucks as opposed to at your house.