If that person is someone you know and see from time to time then there is a chance that a rivalry could be instigated - especially if they notice that you are level-pegging with them or bettering them.
Rivalries can definitely be beneficial:
1. You Push Yourself Harder
The main benefit (in my opinion) that a person will receive from any rivalry is that they will try harder and push themselves to make sure that they are better than their rival. If you are in competition with someone and are trying to keep up or better them, then it is only natural for you to push yourself harder and to try and take yourself to the next level.
Don't think that you will be the only one in the rivalry to benefit from this. Your rival will also want to try and beat you so will in turn also make sure that they are more serious about pushing theirself.
2. Motivation
We all become unmotivated at some point or another and find it easier to have a 'day-off' and take it easy. Another benefit that you can see about having a rivalry is that it can help keep you motivated. As your rival is most likely going to want to be better than you, you can be sure that they are going to be trying hard to make sure that they do become superior to you. Having this in mind can help to drive you forward and keep you going so that you don't let your rival get ahead of you.
In the past when I wasn't feeling very motivated to push myself, and I had a rival, I would think of them and reason with myself that they will be trying hard to get better than me whether I'm having a day-off or not - and this would help drive me onward even when I really didn't feel like continuing. Rivalry is one of those few things that can really ramp-up your motivation and help propel you towards your goal.
3. Speed of Improvement
This benefit is conditional on whether or not you are receiving benefit #1 (pushing yourself), but chances are; like in most rivalries, you are. As rivalries will cause a vast majority of people to try harder and push themselves that much more than they normally would, in turn they are generally going to see a greater rate of improvement because they are consciously trying to make a greater level of improvement than their rival is.
This benefit is often unmentioned when rivalry is spoken about. You normally will hear about how a rivalry will cause you to push yourself harder than you normally would, but often because of that benefit you are going to increase the rate of your improvement. It is something that should be kept in mind.
4. Gets you Used to Failure
In an intense rivalry with someone who is on equal-footing with you, you aren't always going to be able to outdo them. You will experience successes and failures both of which are good for you to get used to, particularly failure if you are someone who is not familiar with the concept.
Failure is a life experience that is good for you to be able to cope with because failure is, if we like it or not, a part of life and is something everybody will experience at one point.
Like with most things, people who have experienced failure(s) will be better at coping with it than people who have always seen success. There's nothing wrong with a person seeing success after success, however, people who never really know what failure is, when they do fail have a much harder time dealing with it. Sometimes it can destroy their aspirations completely and really have a negative effect on them - but as is life and we must get used to what it throws at us.
Conclusion
Rivalry can be a very intense and serious thing, but also a very rewarding and beneficial thing if approached with the right attitude. A rival can become someone who you can trust not to go easy on you, someone who will help to make you better through you trying to better them. Maybe over time you will gain a mutual respect for one-another.
"The pressure people put on themselves and the rivalry between the teams is much more marked. And I think that's a good thing. As long as that rivalry remains within the spirit of competition, it can only spur everyone on."
- Eric Cantona