Is it one of those plastic 1/2 gallon death traps? those are one of the worst possible environments for your bettas.
If one is sick, they might both get sick because they share the water.
Can you possibly get them into bigger tanks with heaters? IMO 2.5 gallons for each betta is the least but anything under 1 gallon for each betta is just ridiculous. They are tropical fish and should have a constant temp of 78-82 degrees F to thrive.
Once you seperate them, heres how you can treat the popeye...
•Symptoms: One or both eyes will swell and bulge out. It in itself can also be a symptom of Dropsy.
•Treatment: Usually not fatal if treated, but your betta might lose an eye. Conservative: If the swelling is extreme then use ES first. Perform 100% daily water changes. You may be able to treat the popeye with just Epsom and clean water. However, if the swelling goes down but the eyes remain cloudy/white, then switch to Aq.Salt at 1tsp/gal or use medications. Do not combine Aq.Salt with medications but you can continue to use ES during medical treatments. Medication: If ES or Aq.Salt do not do the trick or it is a serious case of popeye, then combine the ES with API Tetracycline, API Fungus Cure, API General Cure, API Triple Sulfa, OR Mardel’s Maracyn.