It makes the soldier take +25% more damage from his/her own attacks. This has effect on weapons that cause explosions, Elementals, and few Experimentals. Having Embarrassed flaw makes suicides easier. This can be fixed with Pride. If you add/unlock that upgrade, the hero will even take -15% damage from own attacks.
Some soldiers can have this flaw, but it won't affect them. Those classes are Ninja, Sniper, Medic and Gunslinger. This is because they don't use weapons that can harm them or their squadmates. However, those soldiers shouldn't fight in gun games unless they have Pride.