Tooth picks work great.
The glue I like best is JB Weld. I mix a little up, roll the toothpick in it and then stick it in the hole. I stick as many round toothpicks as will fit in the hole ( I break the first toothpick in half and the first one fat side down then the rest with the tapered ends down.)
After its dried I predrill a pilot hole for the screw, if you put the screw in while the glue is still wet you wouldn't need to pre drill. I have yet to have a screw pull out doing it that way. And it's something that is easy to do while traveling.
Clean off any excess JB Weld before it drys, it easier then snading it later.
Never done a screen door.