If a company is listed on two exchanges, say, with share prices in both USD and EUR, are the prices kept in synch with each other, or allowed to float independently? If the prices are coordinated, how is the calculation made to make sure they are in synch?
The prices are not correlated by the exchanges, but there a host of arbitragers out there who will immediately pounce if the prices drift out of line by the slightest fraction, buying on the cheaper exchange while simultaneously selling on the more expensive exchange, and in practice this keeps the prices the same.
That's exactly as Mike said - the price is not coordinated whatsoever. It's the invisible hand of the market (arbitrageurs) that keeps the corresponding prices at a rationally equal level.
You have to know though that this works that way if the assumption of full fungibility holds true (shares can be easily exchanged/converted between exchanges).
Most of the time you're not directly purchasing foreign stock, but using Depositary Receipts (most commonly ADRs). A more thorough explanation can be found for example in a monograph by Karolyi (1998). The thing is, they are not 100% equal to the underlying stock, especially if you take into consideration all additional costs and limitations to get ahold of the stock in its home market.
Nevertheless, most of the research studies seem to confirm that there are no significant arbitrage opportunities in the ADR market, and if it exhibits large premiums, then they dissipate quickly (Gagnon, 2004).
NYSE Global Shares® are another type of cross-listing where you have a real "global stock". This is a very different scenario. So in this case the price is "coordinated" - you're directly trading ordinary shares of a foreign company in a global register.
Last thing to note, cross-listed securities shouldn't be confused with dual-listed companies, which are a result of an equalization agreement and are completely different breed. In that case, despite being economically equivalent (perfect substitutes), stocks and shareholders of the twin companies are separate, and arbitrage is not always possible in a timely manner. Thus important mispricing may occur in such situations (Jong et al, 2003).