Each custom asset has two components, an 'asset package' (.assetpkg) and the 'generated asset' itself, which consists of a folder that holds a number of subfolders and files.
The asset packages will be saved in your 'asset projects' folder and the generated assets in the 'assets' folder. The folders are customizable so you need to check your settings to find out where they are being saved on your machine.
If you want to share the asset you have 3 options:
1) share only the 'generated asset' (for example by making a zip file of the folder holding all the generated files). This will allow the user to paste that folder into their 'assets' folder and it will appear in their library, but they will not have an option to customize it (change the category, tweak the materials, modify the name...)
2) share the 'asset project'. Now the user will have the option to tweak the asset, but it will not immediately appear in their asset library. They will first need to open the asset project in the asset editor, and generate the asset from the project.
3) share both the 'generated asset' and the 'asset project'. Here the user gets a customizable asset but can immediately start using it without having to generate the asset first. The downside here is that you have to make sure the user drops both the 'generated asset' and the 'asset project' in the correct folders.