Still can't find the exact article I was referring to, and other research indicates that you are correct stating that there is no specific prohibition on private claims.
However, this is an excerpt from the International Convention on Salvage, 1989
(London, 28 April 1989)
Article 13
Criteria for fixing the reward
1. The reward shall be fixed with a view to encouraging salvage operations, taking into account the following criteria without regard to the order in which they are presented below:
(a) the salved value of the vessel and other property;
(b) the skill and efforts of the salvors in preventing or minimizing damage to the environment;
(c) the measure of success obtained by the salvor;
(d) the nature and degree of the danger;
(e) the skill and efforts of the salvors in salving the vessel, other property and life;
(f) the time used and expenses and losses incurred by the salvors;
(g) the risk of liability and other risks run by the salvors or their equipment;
(h) the promptness of the services rendered;
(i) the availability and use of vessels or other equipment intended for salvage operations;
(j) the state of readiness and efficiency of the salvor's equipment and the value thereof.
Again, not an attorney, nor did I stay at a Holiday Inn, but it would seem that (i) would preclude most private vessels from just popping up on the scene and then claiming salvage. i.e. putting out a fire with their saltwater washdown and specifying it as "equipment intended for salvage operations."
Link to complete convention:
http://www.admiraltylawguide.com/conven/salvage1989.html