System Information
Operating system: Windows-7-6.1.7601-SP1 64 Bits
Graphics card: GeForce GTX 770/PCIe/SSE2 NVIDIA Corporation 4.5.0 NVIDIA 430.64

Blender Version
Broken: version: 2.80 (sub 74), branch: master, commit date: 2019-07-12 17:50, hash: rBe3c586e262dd
Worked: (optional)

If for whatever reason, you have multiple rig objects on a mesh as multiple Armature Modifiers, It will be continuously difficult to select bones between the armatures.
When selecting a bone, it will remove the other armature from pose mode select.
So, rather than just clicking the bone, you have to

1. click the bone (to make it's armature active), 2) click the mesh to go back to weight paint, 3) Click the bone again to set it's vertex group as active.

This only applies to the armature modifier(s) that isn't the last one in the stack.

This leads to another issue if you use Bone-Xray.
I get that you can't select a mesh from pose-mode with xray but since the rigs aren't working together here, there becomes conflicts over bone xray here.
To bypass this, you have to 1) disable xray, 2) select a bone, 3) select the mesh, 4) select a bone again, 5) re-enable xray, then 6) select the bone you want if you couldn't select it before.