This will probably be answered by personal experience rather than
through research. I'm going on a business trip to Talinn, Estonia next
month. While the official currency is the Estonian Kroon, the hotel
I'm staying in billed me in Euro, and various accommodation websites
I've seen also quote in Euro. Does this mean that Euro are widely
accepted in the city, thus saving me a trip to the bureau de change,
or do I need to order Kroon?

This site: http://www.virtualtourist.com/f/p/5e18/ states that shops
don't accept foreign currency, but that there are plenty of currency
exchanges around, so you can bring $ or Euros and exchange them fairly
easily there if you need to.  I guess it would depend on whether you
were staying in a larger city or more remote areas.

As far as I know, one can pay only with Estonian currency in Estonia.
Accepting of foreign currencies is prohibited. Nevertheless, you can
exchange your money in Tallin quite easily.