Here's what occasions my question: I submitted an answer (the first answer) and in the course of the next 24 hours three other users provided essentially the same answer. Until now, my conduct has been to hold back posting an answer when I see that this answer has already been submitted? What is the community protocol on this?
In the big picture, answers that merely repeat existing answers aren't good for EL&U. Intentionally duplicating someone else's answer and relying on a higher site reputation score to poach votes from the earlier poster would seriously transgress against the site's rules and against the collegiality of this site. I don't think I've ever seen it happen here, though it might on very rare occasions.
My sense is that the repetitions that do occur are almost always accidental: A person doesn't read the earlier answers closely enough (or at all) or somehow doesn't recognize the great similarity between the existing answer and the newly submitted one. After all, it's not as though duplicate answerers are trying to get away with plagiarism; anyone can see that the other answer says the same thing and was posted earlier.
In my experience duplications occur most frequently in single-word request answers, where a finite number of relevant short answers are possible, and where multiple people think of the same word and then scan down the list of answers to see if anyone else has already posted that suggestion. It's easy to overlook an earlier suggestion of the same word in that situation.
I'm usually pretty good about not offering answers that significantly overlap other answers, but I blundered into that situation within the past 48 hours, just because I wasn't careful enough in reading a response that someone had already posted. If the author of that earlier answer hadn't pointed out to me the overlap in our answers, I still wouldn't be aware of it, so I'm grateful for the notification I received. The author of that post was a good sport about the whole thing, and I decided that there was enough non-overlap in our answers to justify me in leaving my answer up rather than deleting it. But the experience served as a reminder that being a good reader of other answers is a skill that requires a person's whole attention.
Pointing out a duplicate answer to a later answerer may encourage that person to pay more attention to existing answers before posting, thereby reducing the number of duplicate answers in future. We all benefit from being more careful in our reading of other posts and in our crafting of our own. Still, I would hesitate to call someone out for an answer that duplicated mine if the duplication were simple and obvious (as when both answers suggest the same single-word answer).
The most effective strategy in that situation (somewhat surprisingly) may be to wait for a third party to point out to the answer duplicator that your answer already gave essentially the same answer. News of the duplication may well come as complete surprise to the later answerer.
When a third party intervenes (whether or not any site participants downvote the later answer for not adding anything new to the discussion), the late answerer is most likely to recognize the disinterestedness of the criticism and to take to heart EL&U's disapproval of needless repetition in submitted answers. At the same time, a third-party intervention helps the person who submitted the original answer avoid sounding unduly proprietary or point-conscious.
Since you don't submit answers that others have already posted, Little Eva, you are already following the appropriate standard, in my opinion. The question of how to deal with situations where someone else duplicates an answer that you posted earlier is harder to answer; but if you start from the assumption that everyone participating at EL&U means well and wants to help questioners find useful answers, I don't think you can go far wrong.
If your answer is not bringing something new to the table, then you aren't doing anybody any favors by attempting to siphon off some rep.
When I see duplicate answers, I will only upvote the one that has the earlier timestamp. Sometimes I will also leave a message to the duplicate-poster (especially if they are new) indicating that an answer should have something to distinguish it from others. Occasionally even a downvote may be in order, or a flag for "should have been a comment" if the post is of the form "I agree with X, who said Y." I believe that this is the generally accepted strategy of the community.
Occasionally, of course, honest duplication happens (often checking the timestamps shows that the answers were posted very close together). If I find myself the duplicator in such a situation, I'll either work to distinguish my answer, or delete it; if I see an honest duplication, I'll probably ignore it.
When I see near identical answers posted, I will leave a comment and direct the answerer to the post that contains the same answer. If the answerer does not edit his or her post, (obviously I wait a bit) then I will flag it. I may even vote to delete it. Two or more identical answers are pointless.
But if the later post contains information, shows some research, has a different angle and/or is obviously superior to the original, then I won't flag it/them. And I probably won't leave a comment either. Let the community decide which answer(s) they prefer.
If I myself have posted an original "answer", and others come along and "steal" my research, copy my ideas, and post them as their own, then that is plagiarism. I don't think this has ever happened to me, but I would feel mighty upset. Ultimately my hope would be that other users leave pertinent comments, flag and/or downvote the offending post.