How can I paint both sides of a door without waiting for each side to dry first?
Remember the saying, "Haste makes waste". I understand the convenience and speed of hanging the doors so you can paint both sides at once with a sprayer. However, if you are looking for a smooth professional finish without runs, orange peel and loss of the millwork detail, painting them while hanging vertically will will not give you good results with standard latex paints. Shooting latex is not like shooting auto lacquer paint.

When painting doors, gravity is your friend, thus they really should be laid down on horses or supports with sharp stand-offs and done one side at a time.  This minimizes dripping and runs and allows paint to level before drying.

Spraying only will give you a bit of an orange peel effect and possibly flood details. Unless you are really good and are using a high quality sprayer with the proper nozzle, the coverage will be uneven. Spraying is a great method of getting a lot of area covered quickly and fairly evenly. Even us pros will overbrush quickly after spraying with a super soft Purdy XTRA-Glide or equivalent brush. This allows you to get into the details, smooth the finish, spread excess paint, even out the coverage and eliminate the orange peel effect.   

When spraying latex enamels, thin them with 15 to 20% Flotrol. This will help the paint spray more smoothly with less jams and spits. When you over brush, it will also help the paint level better and give you more working time before it starts to tack.

You can still hang the doors to dry after doing one side. this will save space. Wait about 15 mins after you over brush to hang them so they won't develop runs. Depending on temp and humidity, you may be able to lay them back down on some sharp pointed supports to do the other side in about 4 hours. 
Depending on whether they are bare wood or pre-primed doors, count on at least two coats. If they are bare wood, you will have to add in a sanding, priming, sand again, then two coats of paint.