Can I be prosecuted in a European country while on vacation?

Depending on the countries involved, both are possible and there are a few other scenarios too.

• You are arrested, prosecuted locally and serve your sentence where you committed your crime. As others have said, local law fully applies, it happens all the time. Your home country should be informed and can at most complain and try to exert some pressure on a political level but you don't have any extraordinary legal protection only because you are a foreign national. For a famous recent case illustrating this, see Amanda Knox.
• You are arrested, prosecuted and sentenced but you are allowed to serve your sentence in your home country. As @MastaBaba pointed out, there are international agreements to organise this on a bilateral basis.
• You are arrested, prosecuted, sentenced and you serve your sentence but it doesn't end there. You can still be deported and/or banned after the end of your sentence. Many European countries have actually introduced or strengthened legislation to this end in the last decade. So it's not either face trial or be deported, if you are really “lucky” it could be both!

• You are arrested and criminal proceedings are initiated but you manage to come back home at some point during the procedure. In Europe, for a simple theft with no prior offence or aggravating circumstances, a jail sentence is unlikely and it is therefore unusual to be detained prior to trial. If you choose to “jump bail” and somehow manage to leave the country, things become a bit complicated. Among the possibilities:

  • The country where you committed your crime seeks your extradition to force you to appear in court and/or execute your sentence (not always possible because some countries never extradite their own citizens). For another prominent example, see Roman Polanski.
  • You are sentenced in absentia. Institutions like the Council of Europe have been pushing against this practice but some countries still do it. You might or might not be entitled to a retrial when you are finally caught but one effect is typically to suspend or extend statutes of limitation so that such a sentence means you will have to look over your shoulder and avoid going back to the country of your crime (or possibly all the countries that have an extradition treaty with that country) for the rest of your life.
  • You could still face prosecution in your home country (see below), although that would seem exceedingly unlikely for a theft.

• Nothing happens, you come back home and are prosecuted in your country of origin. Many countries do that for particularly serious crimes but some countries also allow it for pretty much everything, in particular because they never extradite their own citizens. In the latter case, the crime in question must typically exist in the laws of both countries whereas in the former case, it's sometimes even possible to prosecute people for things that were not illegal where and when they happened (e.g. in many European countries, laws against child prostitution typically apply to offences committed abroad, without any reference to local law).

• You are prosecuted in another, unrelated third country. It's unusual and typically limited to particularly heinous crimes (and in particular war crimes, crimes against humanity and genocide) but it does happen. Among those that might take interest are your country or countries of residence or the country of origin or residence of any of your victims and Baltasar Garzón. See also Wikipedia.

Among other curiosities, the fact that it might sometimes be possible to be prosecuted in several places also raises questions related to the application of the non bis in idem principle.

In short, it depends a lot on the details, there is no overarching principle or international agreement that would apply in all cases.

It would depend on the crime and sentence, but basically:

The US cannot get a citizen out of another country's prison is one example of an embassy stating this, and indeed, especially if they are allies, countries tend not to get engaged in the laws and cases in other countries.

Of course, you can ask your embassy /consulate for a visit and they can do everything to help you in the process.

Sometimes, when the laws differ greatly from your home, your country might offer a trade, embargoes or condemnation - for example, Australia has been quite vocal recently over the impending execution of Australian citizens in Bali. In this case, an offer was even made for the sentenced to do the remainder of their term in an Australian prison, so that's an option sometimes (very rarely though).

You could certainly face trial in the country of the incident though, not your home country, depending on agreements between the two.

Plenty of countries have agreements where you can serve your sentence in your home country after you have been convicted in the country you were prosecuted in.

Being put on trial in another country from where the crime was allegedly committed is extremely uncommon (and also would generally no make much sense), but theoretically possible.

That said, specifically the US has made certain actions a US crime when committed abroad. This won't apply to theft (unless you perhaps plan on stealing the Mona Lisa, though then the US might extradite you to France if you get caught in the US, while France would obviously prosecute), but typically applies to moral and sexual crimes which might not be clearly regulated in the country the crime was committed. Here's a blog that talks about a US Marine who molested girls in Cambodia and was convicted in the US:

http://prawfsblawg.blogs.com/prawfsblawg/2014/06/how-to-prosecute-crimes-committed-abroad.html

Probably the only time you can expect your home country to get involved is when the crime you're accused of isn't a crime in your home country. For example, printing negative opinions about the government isn't a crime in the US, and they might help someone who was arrested for that in a country where it is. If you're accused of assaulting or robbing someone, you might get some consular assistance but don't expect to be immune from the laws of the country you're visiting.

Being deported to serve your sentence back at home is rare and needs a lot of diplomatic intervention. A lot. Being deported to be tried back home? Not going to happen. And most certainly not for stealing.