You shouldn't try to "out do" the loud davener since that would just disturb even more people.
Re how to correct the person. This is the category of
rebuke that is mentioned in Lev 19:17. The Rabbis and the sources spend considerable time examining how to properly rebuke a person.
Some ideas in reference to your situation:
- I'd do it privately with the person afterwards. No one should be able to hear or realize that you're talking with the person about the issue.
- Get to know the person first--do you know his name, his circumstances?
- Be very gentle--"did you realize that you were saying the silent Amidah loudly and it might have disturbed others' prayer?"
- One of your goals should be to continue the relationship after the rebuke. In other words if your rebuke leaves the two of you not wanting to subsequently talk with each other, then you have failed in your rebuke.
Related to the last point: Most importantly If you are not sure that you will still have a relationship with the person after the rebuke then it is better to not issue the rebuke. (Properly making this judgement requires that you really know the person before considering to rebuke or not.
Remember that you are rebuking the person to help them onto a better path. If you suspect that the person won't "hear" you then you should not deliver the rebuke. You are not to give a rebuke just so you feel better / more rightous / whatever about yourself.
The issue is to assist the other person....
Sources: my teachers. There are many famous gemara on this issue starting with Arachin 16b. A blog post.