You can save styles and share them... don't know about SDE
To save a style to share, you just need to create a style in a project and populate it with symbols or other style items. You choose a file location for the style file when you create it. This .stylx file can then be shared the same way you share any other file.
Then there sharing for the web
There are many other links on 'sharing' within the same vicinity in the help links above.