Firstly, I always decorate first on plain glass and get it silvered later. Trying to remove sections of the silver backing from an existing mirror is working backwards in my opinion. If you do wish to decorate an existing mirror, the backing paint must be removed from that area, then the silver removed with acid.
The usual paints used on glass, alkyd enamels and other oil-based finishes, are unreliable under a mirror backing. The silvering process is harsh on paints with both mechanical damage during thorough cleaning of the glass and chemical damage due to the strong caustic solutions in a hot environment.
Silvering occurs with the reduction of a silver salt (usually nitrate) causing the silver metal to deposit out onto the glass. The silver is traditionally backed with a red lead backing paint thinned with xylene. This is also a threat to some paints because xylene is a moderately strong solvent that will swell and wrinkle fresh enamel paint if it hasn't fully cured.
Ammonia and hot water are used which is a problem for alkyd paints. Oil and alkyd paints contain many ester linkages. Esters are saponified with alkali back to their original acids and alcohols. Under the caustic silvering process, these paints break down leading to either lifting of the paint film, or a hazing effect, like a halo where there should be a clean edge.
The traditional method used in decorating mirrors is to protect the painted work with a bituminous backing prior to silvering. I sometimes use Black Japan by Feast Watson. It is a dark brown (virtually black) translucent coating that is thermoplastic (not thermo-setting like standard paint) and will dissolve with turpentine but is resistant to silvering. The manufacturer calls the resin "Gilsonite", but I assume this is the same as the paint known in USA as asphaltum.
The problem with this approach is that the design has to be backed up accurately right to the edge to be effective. This can be especially tedious with lettering.
Epoxy finishes are good under silvering. Production mirrors are often screen-printed and the appropriate epoxy screen inks are suitable for brush application too. I have used Sericol Polyscreen PS, an epoxy 2-part ink that meets all the requirements. It has good adhesion to glass and is compatible with silvering. I found it works OK in the brush though it doesn't set up like enamels and tends to sag for some time, it air-drys in 1-2 hours. It must be fully cured which may take several days, or it can be low-baked. It can also be used as a backing to enamels as the thinner doesn't attack oil-based finishes.
A different approach is to masking off areas of glass to protect it from silvering. It can later be filled in with the painted design. I have tested a variety of films, tapes and masking paints. All the vinyl films worked without any problem. They gave a good edge and peeled out easily. The metallic mylar films, masking fluid, cellulose tape and masking tapes aren't recommended. They either allow some seepage at the edge or fail completely (in the case of cellulose).
Best results are obtained if the masking is peeled out when the backing paint is still fresh and soft. Water based coatings and latexes are not suitable for masking.