Sounds like the latex paint is not compatible with the previously stained finish or the door wasn't properly prepared for the fresh coat of paint. In either case, you probably aren't going to like our answer, but we hope that you will see the value in the long run.

For a long lasting "non-stick" finish we suggest that you start from scratch. Sand off the gloss exterior latex paint finish right down to the previously stained surface. Apply a coat of high quality oil base primer and touch sand it smooth when dry. Vacuum the dust and apply two thin coats of high quality oil base enamel in the luster of your choice.

Although you could again use latex paint, we suggest that you use an oil base finish. Oil base paint is more abrasion resistant, is easier to keep clean and will cling to your entry door like nobodies business.

While we prefer oil base for all interior doors and trim, you can get away using latex provided that the surface is properly prepared and you use good quality paint. The fact that it is sticking leads us to believe that the quality of the paint may be a question or it was applied too thick. In either case, a light sanding a application of a thin coat of finish should do the trick. Be certain that none of the doors are closed until the paint has had an opportunity to fully dry -- usually at least a week. This may mean removing the weather stripping until the paint has dried.