Running water in master shower causes guest bathtub spigot to drip
I have seen this with washer less faucets. The pressure drop causes them to drip especially if they are older. A set of new seals and possibly springs may tighten it enough to stop the drip, The big box stores Cary kits with different sizes some times they are color coded to size, there are at least 3 different sizes that I can remember, or you could pull the cartridge and take it with you to the store to make sure to get the correct size
here is a sample of what I think you need.