To share something without giving it away physically?

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?

I disagree with your premise. To share does not mean, necessarily, to lose something.

Let's consult the Wikipedia page on Sharing:

Sharing is the joint use of a resource or space.

And of course, in recent times, the term "P2P sharing" has been popularized.

Peer-to-peer file sharing is the distribution and sharing of digital media using peer-to-peer (P2P) networking technology.

Even before there was any of this technology, individuals still shared stories and shared a good laugh—all without having divided one material resource and transferring it to another.

So, the verb that describes sharing something without losing it...is sharing.

For the context involving information and knowledge, you could consider using disseminate:

Spread (something, especially information) widely: health authorities should foster good practice by disseminating information

[Oxford Online Dictionary]

In terms of music to share might be better than to disseminate and the linked Ngram Viewer shows the difference in their usage.

In terms of image, it is idiomatic to use to share and it doesn't necessarily mean that you lose it.

May be not the exact answer, but inspirational.

Socialize

From Wikipedia :

Socialization, also spelled socialisation, is a term used by sociologists, social psychologists, anthropologists, political scientists, and educationalists to refer to the lifelong process of inheriting and disseminating norms, customs, and ideologies, providing an individual with the skills and habits necessary for participating within their own society. Socialization is thus "the means by which social and cultural continuity are attained".

and from M-W Socialize means: "to talk to and do things with other people in a friendly way"

However it has different meanings in different contexts, as in following examples it introduces a sense of sharing something with others.

When you socialize your idea with your friends (and eventual backers), they may say, “Are you kidding? That $100,000 is worth more than 20 percent.

and

Socialize your information and recommendation with a variety of people to verify the reasonableness of your analysis.

and another one

You can also socialize your images on different social networks. It does comes with some problems, like, there is no option of deleting an image...