If a man needs to be brought to the hospital on Shabbat due to a life threatening situation, another man may bring him there even by violating biblical Shabbat prohibitions (e.g. driving a car). Which prohibitions (such as driving or Techum Shabbos) can the second man violate to return home on Shabbat, if any?
I too must echo YS that this must be discussed with one's own halachic authority if it is a practical concern (as my teacher instructed us when teaching this topic).
Rav Moshe Feinstein zt'l ruled that one may return in order that one is able to assist the next time (I have not studied this inside but a quick glance and the secondary literature would seem to suggest that this leniency might be limited to professionals). Igros Moshe O.C. 4:80
Rabbi Ostroff has said that Rabbi Shlomo Zalman Aurbauch zt'l prohibited one from violating a Torah prohibition to return from an emergency call on Shabbos, but permitted violating a Rabbinic prohibition (i.e. instructing a non-Jew to drive oneself home). It would seem that Rav Yitzchak Yosef shlita also prohibits driving home oneself (Yalkut Yosef 328 page 208, cited in Practical Laws of Shabbat Part 2, Rabbi Rafael Cohen Soae, page 224).
One who has gone to retrieve a doctor may return with him to help show the most direct way and/or otherwise assist (Yalkut Yosef 328 page 210).
I would think that it would be forbidden in most circumstance, but I thought of one that might permit it. Here goes. Would it make a difference if the person driving the choleh to the hospital would otherwise not agree to drive him unless it was on condition that he could get a heter to drive back home afterwards? I think that if so, and assuming you couldn't find anyone else to drive the choleh, it would be permissible to let him drive back because otherwise there is a pikuach nefesh problem and he would die, ch"v.
To give a real example of the situation that Adam Moshe raised as a theoretical possibility:
I was not raised Orthodox (what would now be considered Traditional Halachic Conservative) but I encountered this situation once in my childhood - my sister was injured on Shabbat and my mother, without hesitation, bundled us all into the car and drove her to the hospital; clearly permissible, as this was the fastest way to get my sister to the hospital.
If it were not permissible to drive all of us home from the hospital (and remember that this was in the days before mobile phones, and even before 911) my mother might have hesitated over the logistics of getting my sister to the hospital while dealing with other small children too young to be left alone. Might have hesitated long enough to have affected my sister's health.
(I believe that it is completely irrelevant to the halachic question that my sister's injury turned out to be far less serious than it looked; I include this fact only to reassure the community that I am not sharing the sensitive story of some horrible tragedy from my past.)
If it is practical to reduce the degree to which you break Shabbat to deal with a medical emergency that should be done, but in practice "take someone to the hospital" may need to include "and bring them home afterwards."
After prompting by helpful commenters below I found this:
Eruvin 45 "ALL WHO GO OUT (beyond the Shabbat limit) [IN ORDER] TO SAVE LIFE MAY RETURN TO THEIR ORIGINAL PLACES." as quoted in talmudtweets.blogspot.com/2013/04/eruvin-45-national-guard.html
That seems to be mostly dealing with midwives carrying the tools of their occupation and militia members who are permitted to carry weapons home after dealing with an incursion (as otherwise clever foes learned to attack militia members walking home empty-handed.)