The gemara[1] does appears to say, as you indicate, that one must own his matza. In discussing matza that is made from ma’aser sheni,[2] the gemara says that according to the opinion that ma’aser sheni is considered “HaShem’s property,” one cannot use it to fulfill the mitzva of eating matza. This is derived from a textual connection between matza and challa[3] taken from dough, which applies only to one’s own dough. We accept the opinion that ma’aser sheni is in fact the property of its human owner, and thus the question is moot in that regard.[4] However, the concept is reflected in the halacha that one does not fulfill the mitzva with stolen matza, which is likely due to a lack of ownership.[5] Therefore, there is a potential problem even if one “steals” matza unintentionally and no one cares about it (e.g., two people mix up their matzot).[6]
Why, then, do we not find that people are careful to make a halachic acquisition (kinyan) of the matza? The lack of concern about this ostensibly important halachic issue is significant on two levels, in this case as well as in many issues in Halacha. First, the fact that there is no minhag to be careful is itself halachically significant. Perhaps more importantly, the classical poskim are silent on the subject of needing to make a kinyan before eating the matza that one is given as a guest. This phenomenon, called setimat haposkim (the silence of the halachic authorities or their not having made distinctions between cases), is a major halachic factor. In other words, if there were indeed a need to do something extra in a common situation, classical poskim would have spoken about it. (When discussing unusual occurrences, it is more reasonable to say that the lack of a source or a minhag of stringency is due to a dearth of discussion about rare cases in general. That cannot be said here, as guests for the Seder have always been common.)
Therefore, we do not suggest that one go out of his way to be stringent and have guests make a deliberate act of kinyan before eating matza. Creating a clearly new chumra is unwarranted and thus has a negative effect. Let us, then, explore why there is no need.
The Sefat Emet[7] suggests that we can apply the Rosh’s position about a groom who “borrows” a ring to effectuate a marriage. We assume that it was given to him to acquire it halachically, for if not, the marriage will not take effect. Similarly, the host gives the matza to his guests as a present, and the guest receives it with that intention. This explanation is somewhat difficult, as many people are not aware that one needs to own the matza, such that the assumption of their intention may be unrealistic.[8]
However, there is a simple answer. Intention to acquire is pertinent when one could be acting either to acquire something or to borrow it. If the object will be returned to its original owner, the transaction is considered borrowing unless one goes out of his way to acquire the object. However, when one receives matza in order to eat it, he will obviously not be returning it. Thus, the scenario itself demonstrates an effective intention to acquire the matza. Since putting something onto (or into) one’s body is an act of kinyan,[9] one acquires the matza by the time he swallows it.[10]
Another possibility is that one acquires the matza when he makes a change in it by chewing it. The mitzva is fulfilled seconds later, when he swallows. This argument does not apply to stolen matza[11] because the beginning of the action in “fulfilling” the mitzva is forbidden; since the matza is acquired by a forbidden act, the entire action is disqualified.[12] This explanation is also difficult because changes to an object affect ownership only in cases such as that of a thief, where the thief previously did an action to try to bring the object into his control, and that element is missing here.[13]
Some poskim disagree with the premise that you arrived at based on the gemara – namely, that one needs to own the matza. Some argue, based on the comparison to challa, that one requires only full permission to eat the food as he likes, but not actual ownership.[14] This certainly exists for guests and household members.
In summary, although it is not clear what the correct halachic explanation is, guests at another’s Seder do not have to do anything special to make sure they own the matza that they will eat.