I know it’s not the pickiest of tests, but I usually deem it sharp enough when I can shave hair off of my hand/arm with it. I suspect the paper cutting test is probably more consistent, and indeed, less sacrificial of one’s body hair.
But nothing is quite as convenient as the shave test.
The true test of whether a plane (or any other edged tool) is sharp enough is whether the tool will perform the intended task. Sometimes fitness for use is a combination of shape and sharpness. My test for fitness varies depending on the tool. A chisel or drawknife normally just goes to the wood to see how it performs while a tool that requires assembly and adjusting, such as spokeshave and plane blades, gets a little test before the work of assembly is started.
I have been doing this a while and have used the shave test many times for straight edged tools. I don’t recommend it for curved edges such as those found on gouges and cambered edges. You can often spot a woodworker in a crowd by the lack of hair on the non-dominant arm. After a while you fall into a sharpening routine and consistently produce a fit edge by simply following that routine. After that you just do the routine and put the tool to work. If there is an issue with how the plane works you may repeat part of the process and then perform some (pick your favorite) test just to verify success.