Our small door opens out to save floor space, and the big French doors open in, because of the roof overhang. (Usually only have the one you see open, but sometimes both.) Makes using screen doors problematical! So far it's not a huge problem with mosquitoes and other bugs indoors, though Joseph got MOBBED on our front porch yesterday.
Considering this solution, though:
hanging screen doors like these from ecobottoms.com.
Or these, from bugsoffscreen:
bugsoffscreen.com
Or from Amazon:
(We can make screens for the windows, they just fit indoors, since we have awning-type openings, instead of out, if need be!)