Hi, I have an Intel based iMac I use from home on a highspeed connection. When I search for something with LW, many of the results I receive show the checkmark that means I've already downloaded the file when in fact I haven't, any ideas?

There are situations where the checkmark will see the file in in your Library, but it has a different name.

So, perhaps you are sharing a file that is actually the same as one in the search results, even if the names are different.

Check the filesize and see if the files that size in your Library are actually the same file.