Question is as in the title:
Is there a way for moderators to contact MO users personally?
Yes, moderators can contact a user that has an account in the community they are moderating using private messages. When the moderator contacts a user, the user can also reply (once) to the moderator within the same interface.
Also, moderators can create private chat rooms that are more suitable for prolonged conversations.
For more details, see this Moderator FAQ entry from the main Meta.
In response to Praphulla's comment below the question: normally moderators are made aware of conflicts because someone has flagged them for attention. A common type of flag presents mods with the ability to delete or edit a comment, or otherwise decline the flag. Most of the time it doesn't rise to the level where a mod sends an explicit message to the account whose behavior catalyzed the flag; usually such moderator messages are reserved for serious issues which may be about ongoing behavior, as opposed to temporary flare-ups. On other occasions, a mod may move a long stream of comments (e.g. arguments) which are not crucial to the post over to 'chat', and delete some or all of such comments under the post.
Just as a reminder: if users want to reach moderators about an issue that is not comfortably handled publicly (e.g., in meta), they can write to [email protected].