You know, I've never looked at the "File > Library > Sync Library" function and didn't know what it did. But it says what it does clearly at the top of the dialogue;
"Library Sync lets you copy files from another PC. Just start Library Server on the other PC, then start the sync here."
That is definitely not the right thing to do, unless you were starting from a clean MC installation on the second PC, and actually wanted to copy all your media files across from an existing PC. That is not the sync that a Client/Server setup uses. But it does use Media Network to read all the files from a Server Library. The "File > Library > Sync Library" function is only available on a Client if the Client isn't already connected to a Server anyway, so normally not an issue. But MikeO gave you a bum steer there.
In fact "Library Sync" sounds a lot like using the "Handheld Sync" function to sync files to a hard drive directory. That process can have some gotchas, so I wouldn't be surprised if this process does as well.
No, if you wanted to set up a Client/Server environment in MC, follow the instructions in the Media Network Wiki article Jim mentioned. Once that is set up and you are working on a Client that is connected to a MC Server, the Sync that MikeO meant to direct you to can be found in almost the same place, at "File > Library > Sync Changes with Library Server", which only syncs changes in the Library itself, and does not copy files. The media files that are located on the MC Server are then served to the MC Client as required, using the Media Network functionality. Sounds confusing as I write that, but it isn't.
The correct sync can also be found by clicking on "Playing Now" then "Playing from..." then the name of the MC Server you are connected to. A button will then be visible to the right titled "Sync Changes with Library Server". Click that to sync just the Library.
If you ran the "Library Sync" function on your second PC it probably didn't copy a bunch of files across if you left the default settings in the dialogue, because the default is to Merge files, by which I assume it means MC won't create duplicates. But if the file paths you use on the two PCs are different, it may have started to copy files across, because files in a different directory are different files are far as MC is concerned, with this function, I suspect, since I haven't and won't be trying it!
If you have a mapped drive from the second PC to the first PC, then Auto Import on the second PC could have started importing files directly from the first PC, if your settings allowed that. But that would have happened previously if both PCs were turned on and on the network.
So you will need to clean up any duplicate files, and perhaps it would be best to restore a recent backup of the library on your second PC in order to get rid of the "Library Sync--Files not in Sync" stuff.
If you do want to maintain a full set of files on both PCs and keep them in sync, then I think you will need to ensure first that the path to the files on each PC is identical, at least below the path defined in the "Options > File Location" settings, and you will need to ensure that the File Locations in the Sync Library dialogue match the "Options > File Location" settings for your local Library. If that was all done, and it may have been when you first ran the function, then the "Library Sync--Files not in Sync" messages probably just mean that the files on the first and second PCs are identical, and so they haven't needed to be synced.
If you do want to do the above I suggest that you set up a small independent library with a few files in a new separate location on the first PC then turn on Media Network for that library, and then set up a new independent library on the second PC, and without copying files beforehand, use the Library Sync function and see what happens. Then edit some stuff on the first PC, adds some files, remove some files, Library Sync and see what happens. You may find it does exactly what you want.
But you really should consider using a proper Client/Server setup. Yes, you do need the Server to always be running when you want to use the Client, but it can be allowed to sleep and then be woken when the Client tries to connect to it. Yes, you do need to do quite a few maintenance functions on the Server PC, such as View maintenance, and any "Rename, Move, & Copy File" functions. But otherwise, as MikeO says, it is a good solution.