Pretty much everywhere you can see a phone number, or perform an action that involves a phone number will notify you if the number you are interacting with is on the DNC list.
For example, when calling via the dialer:
Or, in your calls list & on the lead details page:
The system does not stop you from sending out texts or making calls via the dialer to numbers on the DNC list, however it's imperative that if you do notice the flag that you double check what you are calling about or what you are sending out from the system.
For example, sending a text message to numbers on the DNC list:
The notes for why someone might be on the DNC list are not visible to non-admins.