I have oftentimes flagged badly and am trying to learn what I should and shouldn't flag. However, when my flag is declined, I sometimes would like to go talk to the moderator who declined my flag and get more information. However, this is kinda hard to do with the current system. I don't know who declined my flag, so I have to go ask all the moderators. Is is possible to add a moderator's name to flags?
The name of the moderator who serviced your flag is private information, for the same reasons that vote information and suspension information is private. You don't need to know who downvoted your post, nor do you need to know who declined your flag.
Because all moderators are expected to handle flags in more or less the same way, it shouldn't matter who declined your flag. It's about the flagged post, not about the people involved, in the same way that votes are about the post, not about the person.
To put it another way, how would you feel if the poster always knew who you were when you flagged one of their posts? Would it have a chilling effect on your flagging efforts?
Note that the result of many actions does have a publicly-viewable moderator name attached to it. If your post was closed, the mod who closed it (along with the names of anyone else who voted to close) are made public when the post closes.
In addition, moderators are accountable to each other (we can see the entire history of every moderator on the site), and to SE corporate, who can audit any moderator's activity in detail, at any time.
If you have a beef with the way a flag is handled, post a question here on meta, citing the flagged post. We can look up the moderator who handled the flag and, while we won't disclose who it is, we will notify them so that they have an opportunity to respond to the meta post, if they wish.