I can't speak for all sports, but football (soccer) had frequent matchs between Warsaw pact countries/clubs and western opponents, particularly in the UEFA sanctioned european competitions. Football is no doubt a faily untypical example though, as it is/was a high profile sport with large entourages and media attention.

Because of this, on the face of things, western teams in the soviet bloc were generally treated courteously. However it was normal for the security services to take great interest in both visiting players and fans from within the Warsaw pact who may be hoping to support them - (particularly the DDR had large numbers of fans who would support Hertha, the Berlin team who had fallen on the west side of the wall, and the extremely popular/successful Bayern Munich). Fans would congregate round the players hotels the night before the games to disturb them, although this is not unusual in european club football even today. Certain players were of more interest than others, and the Stasi seemed to be the security agency most interested in linking football across the iron curtain to their work - the Stasi museum in Berlin has a huge section dedicated to their actions in this area. Franz Beckenbauers files were huge, if only due to the amount of fanmail intercepted from DDR residents.

One of the biggest effects of playing in the Warsaw Pact countries was undoubtedly preperation. Scouting was very difficult, and in the 80s a problem surfaced in that Soviet players would put in lacklustre performances week in week out in their corrupt national leagues, then play out of their skins on the big stage against western opposition, hoping to earn a contract in the West. Soviet advantage in this area was often aided by the state, even if indirectly - many of the more successful clubs were linked to arms of the state. Dynamo Moscow were linked to the KGB, and their then-director (and current president of the Russian Football Union) Nikolai Tolsykh spent much of the 80s visiting english football grounds. Dynamo Kyiv were the only non Moscow team to challenge domestically, and as such were seen by Ukrainian SSR party members as an unoffical national team.

I'm not aware of too many english language sources which cover this extensively, although there are a couple of good essays by Simon Kuper, particularly in his collection 'Football Against the Enemy' which cover exactly this - check out his piece on Helmut Klopfleisch, 'The Football Dissident' about a fan followed all over by the Stasi due to his obsession with bundesliga and 'connections' to Franz Beckenbauer, and 'The Secret Police Chief at Left Half'/'Rulers of the Ukraine' for the Soviet leagues.