I'm an American citizen visiting Denmark. As I understand it, I can stay in Denmark for a period of 3 months. I arrived in Madrid on May 15th and came to Denmark on the 19th, though my passport wasn't stamped there. I've also been to Amsterdam and came back to Denmark on July 12th, but didn't have my passport stamped there either. Does the trip to the Netherlands count as leaving the country and therefore reset the 3 month period or am I required to return to the States and come back if I wish to stay longer? In case I do need to return, do I have to go on the 15th or the 19th? I also have a Polish passport through which I can stay in the EU indefinitely. Is it possible to have my American passport "stamped out" somewhere and switch the Polish one, thereby allowing me to stay? I look forward to hearing from you.
Almost all countries within mainland Europe (including Spain, Denmark and the Netherlands - as well as Poland) are in the "Schengen Zone", which does away with internal borders. You'd have gotten a Schengen Visa on arrival (on the 15th of May). That allows you to stay within the entire Schengen Zone for 90 days out of any given 180 day period (rolling window, so doesn't reset on day 181). So under those rules, you'd have to leave the Schengen Zone no later than August 12th (90 days, not 3 months).
On your Polish passport you can indeed stay indefinitely. I don't think you'd even need to be "stamped out" or anything. I would expect that you can just show your Polish passport when you leave in case an immigration officer questions how long you've been in Europe. However, this is just what I personally think would "make sense", and unfortunately matters of immigration don't always make sense, so best to find some confirmation of that. (Maybe call up some embassy or immigration department?)
[ Edit: Edited on 31-Jul-2012, at 12:54 by Sander ]
I think the only time on this trip you would really need to use your US passport is on entering / leaving the US. For the rest, I'm quite sure you would have been better off simply using your Polish passport all along. Even at this stage, I'd very much doubt it would be a problem to just leave with your Polish passport and ignore your US one. You are a Polish citizen after all and have every right to stay as long as you want.
Getting confirmation on correct procedures is probably a good idea though - perhaps contact the closest US and/or Polish embassy to ask for their advice?