Well it seems that twitter is one of the more common social networks that people are currently using. It is most possibly at least for me one of the more frequent social interactions that i have. While i work, and go out for coffee, and do any number of other things, twitter is one of the easiest ways to just talk to people at literally any time of day.
While it is a social network, and good for getting connected with people, it can be difficult to track the progress of a specific connection over the long term. For example maybe you are speaking to different people every day, but you find you also seem to have on going conversations with a few people. While it may be possible to just continue on and not really consider this long term conversations, they can almost be one of the most valuable parts of the use of twitter.
I think that there are at least two possible solutions.
The first option is to keep a small journal of people that you have had contact with on networks like twitter. Username, how you first connected, common interests, and just general topics you have discussed.
The second option is to build on the first and actually keep a record of your conversations. This can be a little more time consuming an not entirely feasible if you are looking at a large number of people that you interact with regularly. But being selective it can help you to build a more connected and useful relationship between both yourself and other users.
Just think of it like sales, if you do a lot of sales and have a few customers that you deal with over and over you would not put it to chance that you will remember the details that you need to. So why would you do it for twitter when it has the potential to be a very useful contact tool for you or your business.