Each context-map is uniquely identified by its URI. If you want others to be able to load your map, you have to publish the map first. After that, you can just pass around the URI (just copy the value of the field "Context-Map URI", the same way you send around URLs of pages on the WWW.
If you want to open a context-map, just paste the URI into the field labeled "Context-Map URI" (see screenshot) and press enter. Alternatively you can press the key combination "Ctrl + O" which opens a small window where you can paste the URI into.