The document stays on the machine you opened it on
There is no code in this application that makes a network request, and you do not have to take our word for it — the package can be taken apart and read, and the licence expressly permits that. Turn on airplane mode and every feature still works. The one exception is the optional timestamp on a digital signature, which asks a public time authority to attest the hour: it sends a hash, never your document, and it does nothing unless you turn it on.