The Rashba's position on שומע כעונה by ספירת העומר emerges from a responsum cited by the Beit Yosef (Orach Chayim 489): when the chazzan recites the brachah and the congregation answers amen, one who then went back and recited the brachah again is the subject of the Rashba's ruling — and the Beit Yosef conveys that the Rashba holds that answering amen alone does not constitute fulfillment of one's obligation if one did not have that intent.
This ruling is codified by the Shulchan Aruch (Orach Chayim 489:3) in the Rema's gloss, which states explicitly in the name of the Rashba that even if one answered amen to the community's brachah, if one had in mind not to fulfill the obligation, one may return and recite the brachah and count that night.
The underlying question of whether שומע כעונה applies to ספירת העומר at all is contested among later authorities: the Mishnah Berurah (489) records one view that the Torah's language of וספרתם לכם — understood to obligate each individual — excludes fulfillment through hearing another count, while other later authorities hold that כוונה to be יוצא through another's counting suffices just as שומע כעונה applies everywhere, with all agreeing the brachah itself may be fulfilled through שומע כעונה.
The Rashba's own derivation from וספרתם לכם, as seen in the Rashba (Menachot 65b), is that the word לכם comes to impose the obligation on every individual rather than on the court alone — a reading that supports the view that each person must count personally, consistent with his ruling that mere amen-answering without intent does not discharge the obligation.