A few questions to narrow things down. Are the same arrows missing or is it random? Number them and see. It could be a fletching contact issue and the nocks just need to be indexed slightly to fix it.
And when you say "shoot through" rest, what rest are we talking about? That leaves a lot of rests to think about, some have particular issues that others don't. It's ok to tell us the make and model of rest you use
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Also bow specs would be nice. Draw length, peak draw weight, length of arrow, model and size of arrow and your tip weight would be nice. Although I doubt it's a spine issue, they would probably still group well if the spine was off, especially with target tips.
And it could very well be a form, grip or shooting technique problem. Maybe this bow is not set up the same as the other one, or your form and grip changed slightly and you don't know it. Inconsistant grip, inconsistant anchor point and dropping your arm will all cause this problem.
Or are you telling us that you own two bows, one shoots great and the other is all over the place? If this is the case it is an equipment issue and needs to be looked at. Two things off the bat would be something is loose, or you have bad fletching contact.
As far as the rest goes, I have just about every type of rest and they all shoot well when set up correctly. If anything some of the drop away rests that have overdraws to them are less forgiving than a decent prong rest. PROVIDING they are set up correctly. Drop aways have timing issues and prong rests have spring tension issues sometimes.
Oh, your arrows could be junk as well unless you are using the same arrow for both bows. Just because they are new doesn't mean anything, especially if they are cheap to mid priced carbons. Spin check those babies and make sure they are straight. Check the nocks and inserts as well when you spin them.
Paul