Question: I thought it was forbidden to transfer ownership between people on
Shabbat. Yet, people regularly bring over food when invited to friends'
houses. Is that permitted? 

Answer: In explaining the prohibition of making donations to the Beit Hamikdash on Shabbat and Yom Tov, the gemara (Beitza 37a) says that it is
included in the prohibition of commerce on these days. Rashi (ad loc.) cites
two reasons for the prohibition of commerce: 1) The navi Yeshaya, taught us
to stay away from weekday-like activities and speech on Shabbat; 2) Commerce could prompt one to write documents.
    Do these concerns apply to giving a present? The Beit Yosef (Orach Chayim 527) cites the Mordechai that one can transfer ownership of a lulav and etrog on Sukkot to one's friend in order to allow him to fulfill the mitzva. The Beit Yosef was surprised with this reasoning, because if one gives a present by handing it to his friend without a formalistic kinyan sudar, it should be permitted even without there being a mitzvah. Some bring proof to the Beit Yosef's own reasoning (see Beit Meir on Even HaEzer, siman 45). However, the consensus of poskim is like the Mordechai, that even though giving presents is not overly commercial in nature, it is included in the prohibition of commerce except for cases of mitzvah, (Magen Avraham 306:15; Mishna Berurah 306:33).
     What, then, constitutes a mitzvah? In addition to necessary religious
articles like an etrog, one may give and receive presents which are needs of
the day (Shmirat Shabbat K'hilchata 29:29). Based on this, people may give
presents of food when going over to friends if they feel that they will be
used to enhance the Shabbat meal (see Shulchan Aruch Harav 306:15 who
corroborates). It is proper to make a realistic appraisal if the specific item given is likely to be used on this Shabbat (for example, giving milchig candy when going for lunch on a "short" Shabbat would be problematic). Giving presents to chatanim or bar-mitzvah boys is discussed by poskim (see discussion in Yechave Da'at III, 21).
     Even when one cannot transfer ownership of an object, one can give it
physically on Shabbat in one of two ways: 1) To transfer ownership to the recipient before Shabbat (Shmirat Shabbat K'hilchata, ibid.), by buying the
object on his behalf or having someone, (preferably outside the giver's family- see details in Shulchan Aruch OC 366:10), pick up the object on his behalf; 2) If the party(ies) have in mind not to affect the transfer of ownership, until after Shabbat.