The links within a test mailing are only active within the most recently-sent email. This means that if you send a test email to yourself, the links will work for you. If you then send a test to another staffer in your office, the links in the test mailer you sent to yourself will stop working and the links in your colleague's email will work since it was the most recently-sent test.
You can send a test to two email addresses at the same time. When sending a test to multiple people, type a comma followed by the second email with no space after the comma so that it looks like this: email@example.com,firstname.lastname@example.org.