In its probably original sense, to share means to divide something among people. In those cases, it is implicit that the sharer is left with less than before:
I had three apples, which I shared with two of my friends. Now I'm left with just one apple.
Another example would be the stock market, where you can essentially buy a part ("share") of a company.
Now, especially with modern technology, it became possible to share immaterial goods where in the end, both (all) parties have a copy of the exact same thing:
People voluntarily share their knowledge on Stack Exchange sites.
In less than an hour, a thousand users shared that image on Facebook.
The fact that people shared their music on the Internet led to a big controversy about copyright law.
In these cases, it is implied that some sort of copying/duplication took place and either way, the sharer has not physically given away what he/she was sharing.
When using divide instead of share, it is implied that the original items, or at least some of them, are given away:
Separate (something) into portions and share out among a number of people:
'Jack divided up the rest of the cash'
In a similar way, is there a verb that can be used instead of to share that explicitly describes sharing something without giving it away?