Bird droppings pose a real hazard to paint, particularly to the newest formulations of water-borne clearcoats.  If left for any time they WILL etch into the surface so the protocol should be to remove them ASAP.

But how to deal with the remaining stain?  The good news: most of the time the stain can be removed using a GOOD quality polishing compound - I do not mean a heavy abrasive that will scratch the paint.  I mean a polish like the 3M Finesse - it polishes (these are professional grade polishes designed for machine use; I use them in seemingly vast quantities and always by hand with perfect results); there are various grades which give the same finished state - only the time required to reach that state is different.  It is good practice to start with the lightest grade possible (in the 3M system it is the "swirl remover") and move to a coarser grade only if results are not quickly forthcoming.  A micro-fibre cloth (which will not scratch) is the best vehicle for the compound.  There are other brands of polishes: Auto-Glym products are generally excellent but in my experience not quite as good as the 3M.  The more commercial brands don't work nearly as well.  The professional-grade polishes are available at shops that supply to auto-body businesses.  These polishes will also remove or lessen scratches in the clearcoat.

*Nota bene: a POLISH is not a WAX*.  If you polish a finish you are smoothing it by removing some of the finish.  The resultant smooth finish MUST be protected by waxing - several coats, preferably (and again, buffed with a micro-fibre cloth) - note also that if you use a "cleaner wax" (the typical commercial wax) then you are using a wax which has an abrasive in it; each time you use it you are removing the previous layer.  In effect you always end up with ONE layer of wax.  A better solution on a smooth finish in good condition is to use a non-cleaner carnauba (that is a non-abrasive) wax such as Mother's (this company produces two types of carnauba: one non-abrasive, and one "cleaner wax" - which cost about half as much per tin.

There are two different formulations of 3M Swirl Remover, one for light paint colors and one for dark paint colors.
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And today's job for me?  To remove hard-water spotting on a recently painted (with a water-borne clearcoat) bonnet (and two days ago I had to remove a bird-dropping stain from the same panel).  I will be using the 3M Swirl Remover and finishing with several hand applied - and I do mean "hand", not cloth or foam-applied) coats of Victoria Wax (the best I have found, having tried everything - and I mean everything except the absurdly-priced waxes made in a small, beautiful and neutral country in Europe.  Somehow the phrase "more money than brains" comes to mind.

I have no affiliation with any of the manufacturers or products mentioned.