Only partially and indirectly known.
W. King(= S. Loyd)
The number of participants is not known, nor if any
honorary awards were given.
White says that the tournament was informal. It seems clear from the requirements that there was no motto requirement, but it is not as clear that submitted problems were published in the column before they were submitted to the judge. The time between announcement (late May?) and the closing date seems to argue against a tournament where problems were published as they were received, although they may have been published after closing date. (Probably requires access to original column to decide.)
White also says that Loyd won both first and second prize in the #3 section. This does not match requirements known so far, which only mention one prize per section. (Did judge recommend adding a 2nd prize for #3?) (The additional problem is on p. 216, prb. 277.)
The prize #3 cited below was later modified by Loyd. The changed version is reprinted in White's book (p. 34, prb. 27), but unfortunately without any indication of where it was first published.
The prize #4 cited below is taken from White's book. It differs from some earlier reprints, also said to be from American Union, as regards the position of the white King, which in those appears on h3. The position given here also appears in Loyd's Chess Strategy, (1878), p. 24, prb. 31, but without any indication that it won a prize.
As little information given here comes directly from American Union, or is contemporary, it should not be regarded as fully reliable.
Section A: #3
Prize: S. Loyd
Key: 1. Kd2
Source: The Era, 21/1049 (1858-10-31), p. 5
Section B: #4
W. King (S. Loyd)
Key: 1. e6
Source: White: Sam Loyd and His Chess problems (1962), p. 168, prb. 196