The proxy can coordinate the communication between parties, perform the initial contact information gathering effort and even, in some cases, conduct its own research on existence, exploitability and impact of the vulnerability. The trusted third party can be of great help if communications between discoverers and vendors are not smooth and cooperative. But the bottom line is that the process of finding bugs, reporting them and getting them fixed can become a very serious and complex matter. All parties have finite resources and therefore it costs them money. Streamlining the process and addressing the problems in a responsible and timely fashion reduces the efforts for all parties.
Communication is key.
The best way to streamline the process and ensure cooperation is to maintain every party informed of what is going on. When that fails, unilateral actions take place that could put all at risk. Conduct all activities bearing in mind that the end goal is to improve the overall security and minimize the harm to all parties. Although this sounds obvious, the ultimate goal can be obscured during the process, evaluate your actions accordingly. Extend the benefit of doubt, do not impute motives.