Mautic’s core function is to track what’s happening on the website(s) where installed the tracking script, and to track how contacts handle emails sent to them (open, click). Mautic organizes this data into processable information helping you to create better, personalized communication with your contacts.
In order to track websites you need to have constant online connection with that website through the tracking script. It means, that every time a visitor comes to the website, Mautic is notified. If the person is recognized, (by cookie or other means) the contact’s profile is enriched with that visiting information. If the person is a new visitor, than a new anonymous contact is created.
With emails it is the same - every time someone opens an email a tracking pixel is fired notifying Mautic, that ‘this person’ opened/clicked an email. This also happens on the public internet.
In other words: your Mautic instance needs to be publicly available on the internet. At least a public IP address, but preferably with a https connection and a real domain or subdomain. (I have a tutorial about how I connected a raspberry pi - pretty much the same story, just with Linux.)
If your question is more about wanting to know if a software is still running after shutting down and un-powering a computer, the answer is philosophical. Have you ever felt, that your PC is trying to take you for a ride? Randomly closing windows? Showing the Blue Death screen when you have important thing to do? Starting an update when you have a zoom call to attend? Did you ever call your PC names? If the answer is yes, you tend to think of your PC as a living thing. If so - the question arises: do living things ever die? Do computers dream? I guess we will never know.
(The author has not slept enough in the past couple of days due to efforts upgrading an outdated Mautic instance.)