There are quite a few private inter-city long distance services springing up. These are run in competition to the existing trains, so they're different than the private regional operators which now commonly compete for government franchises.
Some services are completely private, though many have a consortium that includes national rail providers from other countries (which in theory will someday be privatized).
Austria:
Westbahn runs Vienna to the Salzburg (and its German suburbs). The SNCF is a share-holder. Also the first major Austrian domestic private inter-city bus operator "Westbus." Started 12/2011.
Czech Republic:
RegioJet as mentioned. Poorly named since they have private
long distance service Prague - Ostrava - Havířov - Žilina, Slovakia as well as winning the first regional franchises in the Czech and Slovak Republics. Intercity trains started 2011.
Germany:
InterConnex runs Leipzig-Berlin-Rostock. By Veolia. Started 2002.
Harz-Berlin-Express is among a few small services from the Eastern Provinces into Berlin. By Veolia. Started 2005.
Vogtland Express is among a few small services from the Eastern Provinces into Berlin. By Trenitalia Group. Started 2005.
Hambürg-Köln_Express runs Hamburg-Cologne. Starts 9/2012.
Alex runs Lindau - Munich - Prague / Hof and Nuremberg - Prague. Was run by Arriva/CD[Czech Railways] but the Arriva par was spun off to Trenitalia group with the DB takeover of Arriva. CD run the Czech portion as they would any other train. Started 2009?
Berlin Night Express as mentioned, runs Berlin - Malmö by ferry. SJ[Swedish Railways] operate the Swedish portion, the German part is private.
Italy:
ÖBB Italia in cooperation with DB and local opeator LeNord. Competes with Trenitalia on the cooridor through the Brenner Pass (Tyrol), Verona and Milan. Started 2010?
Thello took over the Venice-Milan-Paris night train. Run by Veolia/Trenitalia. Started 12/2011.
Italo as mentioned. Starts early 2012.
Sweden:
Veolia Malmö - Stockholm as mentioned.
Blataget Malmö - Gothenburg as mentioned.
DSB First Malmö - Gothenburg - Copenhagen, was run under a rare intercity francise by Danish Railways and First (the UK coach operator). Though, I believe, First was recently kicked out of this group, so it's no longer really private.
Plus of course Thalys, Eurostar, etc. As well as some night trains, CityNightLine? And of course tourist and luxary trains.
I'm really curious to see when a private operator breaks into the French market, since walk-up fares are very high. The first domestic coach service just started this fall.