Taking the question literally, you'll find both variants in commercial PCBs, and I'm sure you already know about. Thus I assume, you actually meaned to ask a slightly different question.
IPC-2221A mentions tenting of vias as a way to protect them against process solutions during soldering and cleaning. It also specifies maximum finished hole diameters of 1.0 respectively 0.65 mm for tented vias.
Halfside or incompletely tented vias are inacceptable, because they can't be suffciently cleaned during PCB processing and are vulnerable to corrosion.
I usually opt for open vias with a reduced soldermask opening. They are available as test points and for prototype hardware fixes.