Weird question...

LOLOL! I'm guessing loud enough to blow your eardrums right out.

I think it will depend on the gun. A .22 and a .45 sound very different when fired.

Yes, they would hear it. A gun, even a .22 makes a very distinctive sound.

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I can't access audio files from here, but you might be able to google ".22 sound effect" and get to hear what a fired weapon actually sounds like. Then try to account for how much the mouth would muffle the noise (my guess would be to a limited extent), imagine it filtered also through a closed door who's nearby, what other sounds are going on (is the hospital really busy? is it dead quiet?) and run with it from there.

A .22 is a difficult round to kill with. It's small and could easily be deflected by bone. The placeing of the round would have to be exact.

I'm not sure the context of your story, but if it were me, I would increase the caliber to at least .32 or bigger.

A .22 is good enough if it is a suicide by someone that knows what they are doing. The issue is not just the caliber though but also the type of gun. If it is a revolver, there is no silencing that. It has to be an automatic.

.22 is small enough that you could use the nipple off a baby bottle on the end of the barrel and it will dramatically reduce the noise. This would work on many automatic .22 pistols or a .22 rifle, but only once. but not for the second shot. The reason is that the noise is mostly air rushing back into the barrel. The nipple is pulled into the barrel when the partial vacuum is created slowing this rush of air.

While it is damned hard to commit suicide with a 2 liter bottle tied on the end of the barrel, that also works once, but will be somewhat less successful for the second shot.

The best place to make sure you died with such a small caliber is through the eye. It will very likely not exit the skull, but instead bounce around a bit scrambling the brain. Death is never 100% even with a larger caliber as some unfortunate people have discovered, but while I don't have the exact statistics, most are successful when they place the barrel in the right spot.

Elegy,
Great advice all around, but you also may want to consult a forensics website. Certain guns can make all kinds of sounds, but if placed in a closed mouth, the sound can be stifled somewhat.

I was one told the difference between an unsilenced and a silenced gun is that unsilnced the whole block hears, while silenced only it's only the whole building. Probably depends a lot on the gun (the advice in question was aimed at noir detective fiction writers) but gunshots in films and on television are a lot quieter than the real thing.