Though my coasters are made on rough, unfinished tiles that are porous, here is an idea to use on the finished and much less expensive tile.
You still need to use the permanent Stayz-On inks. It is definitely a little more tricky because the finished tiles are little easier to slip/slide on. I recommend inking the stamp and placing it rubber side up and then pressing the tile into the stamp [generally opposite of what you normally do]. But everyone has to find their own groove. You want to leave at least 10-20 minutes for the ink to dry before handling it, in case you haven't used it in this manner before. The other thing is because it is generally smooth [whether matte or shiny], the ink is easy to scratch off. I recommend a triple thick clear glaze to either spray or paint over it. Available generally in the more professional paint areas of craft stores, though a bit pricey, it will preserve your work better than general clear spray paint. This also helps protest it from not only rubbing off but being wiped away if water/cleaner is used. I always have a note NOT to use water to clean. Though the ink is stated as permanent, it will wash away with water...thus a heavy duty sealant is necessary!
Featured in these pics are my 2 X 2 tile magnets and 6 X 6 trivets. I buy the heavy duty magnets so they really hold stuff well. The trivets are finished with thin foam squares on the back to protect the surface it is placed on. For your info, I sell the trivets for $7 and the tiles for $2.