The "Share" button that you see in both the Dashboard and Search menus refers to the ability for other users in your Organisation to see Dashboards and Saved Searches (e.g. visualisations) that you have created. Once Shared, these items will appear in the "Shared" section of the relevant dropdown menu.
At this point in time, it is not possible to share the dashboards publicly (that is, to someone without a login), primarily for security reasons. However, giving a new user limited access to see dashboards and searches is very easy. Open the Admin menu and click Users. Here you can send out invites to new users and set their permissions to "read only", so that they are restricted to only viewing existing Dashboards and Saved Searches.