You can add instances of a module via the control panel at the top. Normally when you want to add a module, you would select the drop down box and choose a module. Above that there are two boxes, one for new module and one for existing module. If you choose existing module check box, then the drop down boxes will change to "page" and "module". Choose the page the module is on, then choose a module from that page, then click add. Each time to do this, the module does not duplicate, it is just referred to by different pages. Editing one instance of the module automatically updates all the other instances of the module.
So for you, if your calendar is on the Home page, you would go to a different page, check the "already existing module" box, choose home page, and choose the module name, click add, and presto you have your "duplicate." If you change the content of the module at any point on any page it appears on, those updates propagate to the other instances. I do believe, however, that some of the module settings do not get "duplicated", so you could place a different container around it or have different cache settings, etc. Just the main content is repeated. I don't have a control panel open in front of me, so the "add existing module" etc may have a little different phrasing, but it's easy enough to spot what I'm talking about.
Rick