As the twelve come up out of the water they are filled with the Spirit. This event was also echoed in the experience of Joseph and Oliver in the beginning of this dispensation:
Joseph Smith-History:73
73 Immediately on our coming up out of the water after we had been baptized, we experienced great and glorious blessings from our Heavenly Father. No sooner had I baptized Oliver Cowdery, than the Holy Ghost fell upon him, and he stood up and prophesied many things which should shortly come to pass. And again, so soon as I had been baptized by him, I also had the spirit of prophecy, when, standing up, I prophesied concerning the rise of this Church, and many other things connected with the Church, and this generation of the children of men. We were filled with the Holy Ghost, and rejoiced in the God of our salvation.
3 Nephi 19:14
14 And behold, they were encircled about as if it were by fire; and it came down from heaven, and the multitude did witness it, and did bear record; and angels did come down out of heaven and did minister unto them.
This image of the effect of the presence of the spirit echoes the experience with the children in 3 Nephi 17:24. The similar experience from the previous day tells us that this experience would be seen in a larger context of the spirit rather than a specific acknowledgement of the role of the twelve. It was a powerful witness of the Spirit, but not a particular witness of the function of these twelve.