Amino acid supplementation very much depends on the reefing methodology adopted. This might be beating a dead horse but animals cannot synthesize all amino acids. Most plants and micro-organisms however can produce the essential amino acids. Quite often, amino acids and vitamins are leached from plant and microbial sources within the aquarium. Therefore the amount of amino acids and vitamins is dependent on the population of plants and microorganisms within the aquarium. Foodstuffs also represent a very large source of amino acids as well. In traditional berlin systems, the lack of feeding (to reduce nitrates), little or no sand, and extreme skimming creates conditions where algal and microorganism growth is very limited. Limited nutrient input also creates conditons whereby any available substrates (amino acids, nitrogen, and phosphorous) for growth are aggressively scavenged by microorganisms. If memory serves, the amino acid supplementation is "surprisingly" popular in Europe where a great many systems are run in the traditional Berlin style.
Amino acid addition can also be used to alter the behavior of coral. The addition of specific amino acids, namely lysine and proline (if memory serves) often induce feeding responses. You could potentially, induce hard to keep creatures' feeding responses, increasing the likelihood that they will accept food provided.
In my opinion, amino acid supplementation is unnecessary if you feed your tank inhabitants, have a reasonable amount of algae, and sufficient surface area to support a decently sized microbial population. Otherwise, the addition of amino acids is just creating a burden on your water quality.