In my treatment of this question, I was of the opinion that either Moshe bowed down to and kissed Yisro, or else that there was mutual bowing and kissing. Rashi brings down a Mechilta on that early pasuk in parshat Yisro with proof that it was Moshe, based on a gezera shava of ish-haIsh, here with Ish leRei'eihu, and there with veHaIsh Moshe.
But in this past week's parsha sheet, Torah Lodaas, we have the following great, and entertaining explanation, from the Cheshek Shlomo:
That is, we know from various midrashim that Moshe was 10 cubits tall. For example, in Nedarim 38a. Therefore, it is obvious who bowed and kissed whom. Moshe, who was 10 amos tall, had to bow (bend) down and kiss Yisro, who was of normal height and thus only 3 amos tall.
I am not sure whether the midrash found in the gemara was intended literally. That is one question. Another good question is whether Cheshek Shlomo understood that midrash literally. Another (and this is somewhat primary), whether he intended this answer as entertaining Purim Torah or as a real answer. If he is citing his father, perhaps it was not Purim Torah. Another question, what was Rabbi Matis Blum's intent in bringing down this question and answer. Does he mean this seriously? As filler? Because it is entertaining? I know of at least one person who told this over in all seriousness.
What are some problems with this? One "problem" is that Rashi purportedly does not subscribe to this idea of Moshe being 10 cubits tall (see here). First, I am not convinced by the proofs that Rashi rejects that midrash; and secondly, the point is to prove the Mechilta's understanding of the pasuk, rather than give an explanation Rashi would agree with.
A more serious issue -- Moshe was not actually 10 amos tall, despite the midrash that says so. (See link above, and scroll up a bit, for reasons to doubt this on a peshat level). The whole thing is ridiculous, to extend this midrash with further details to answer this question. (The midrash statig Moshe's height also gives an impossible height for Og.)
And even if Moshe were so tall, that is not the plain meaning of the bowing and kissing. The bowing was not bending over to accomplish the kissing. Indeed, we expect vayishtachu to mean that he prostrated himself on the ground, not that he leaned over. (See Megillah daf 22.) And this bowing was part of the greeting. See how Yaakov greets Esav in Bereishit 33, first with repeated prostrations on the ground and finally with a kiss.
All this is not peshat, and even though Rashi is citing a Mechilta to acheive his aims, Rashi wants to derive peshat, from analysis of textual features, not by stretches / kvetches of midrashim.