Operating a flex owned by an operator in a foreign country

Their call sign / your callsign

I have done it both ways many times. 

If the local control operator is in "Control" then the same 3rd party rules apply as the rules for Phone Patches http://www.arrl.org/phone-patch-guidelines .



If there is no "control" operator. then it depends on the reciprocal licensing agreements between countries...

That said, if you are dealing with a CEPT (European Conference of Postal & Telecommunications Administrations) country and the USA you definitely can do it and I have done it many times

http://www.arrl.org/cept


BTW... don't get too bothered by the rules police.. most countries do not care that much about ham radio these days... so enforcement of the rules can be very lax EXCEPT when you do something really obviously against the local rules like "Politics, Profanity, Interference. or beating the locals in a contest, hi Hi,

For instance in Germany the rule would be   dl9ndw/W0XYZ . if you operate a station which is owned and at the location of  another operator, you prepend his callsign . You cannot use a CEPT prepend permanently, just a few weeks and then you need a local license. The duration depends on local rules.  I think in a near future, newer rules will usually cover remote operation which in many countries is still a "grey zone".