Can someone living outside of Israel ask a visitor from Israel to do a melacha for them on the second day of Yom Tov if the visiting Israeli is only keeping one day of Yom Tov?
The main source is the shulchan aruch, 496:3, and commentaries, where a ben eretz yisroel who is traveling in chu"l is forbidden to do melacha.
The Ta"z permits such a person to do melacha in private, only. Most other poskim (including the magen avraham and the mishna brurah) forbid even this. The shulchan aruch harav is also strict on not doing any melacha (although his reasoning is kabbalistic in nature)
Many Israelis, however, do melacha in public. I have not found a source permitting this, however, I'm reluctant to call "everyone" wrong. I have been told (third-hand) that R' Goren permitted this, but I have not seen this in any of his writings (although I admit that I haven't looked).
If the Israeli is in an area where there are no other (non-Israeli) Jews, however, he may do melacha, even publically. Similarly, a traveler who leaves E"Y on Yom Tov Sheni
Note that the opposite case, a visitor from chu"l who is keeping two days who wants to ask his Israeli host to do melacha, has a different halacha... but that should get its own question!
According to the Ta"z, does "in private" mean "not in the company of any local Jews" or just "inside a house"? If it's the latter, according to him, can a local Jew ask a visiting Israeli to do something private, like turning on the bathroom light?– Isaac Moses ♦Mar 3, 2010 at 20:07
He writes "b'chadrei chadarim," if I recall correctly. His severa is the same as the other poskim's; we do not want the Israeli violating lo sisgodedu by demonstrating two different minhagim in one place. It would seem, therefore, that the classification will depend on the outcome, not on the situation at the time of the action. Mar 4, 2010 at 4:22
"If the Israeli is in an area where there are no other (non-Israeli) Jews, however, he may do melacha, even publically. Similarly, a traveler who leaves E"Y on Yom Tov Sheni." How so? If לֹא תִתְגֹּדְדוּ applied in the sense of Y"T Sheni, it would apply evenly to all of the diaspora, would it not (isn't that the point)?– Seth JMar 12, 2012 at 17:34