Like any other occupation, you can learn what you need to know through "real world" practice and experience. That said, the faster and easier way to learn engineering's underlying fundamentals is through a college degree program. A degree is certainly no guarantee that you know your stuff, but it's taken as a strong indicator by employers who often make it a job requirement.
To be an "official" engineer -- at least in the eyes of the government -- you need to acquire a professional engineering license (PE). The license requirements vary some from state to state, but generally require an applicant to:
- pass an engineer in training exam (EIT), which is a test on fundamental theory material like physics, thermal, chemistry, etc..
- acquire engineering-related work experience and/or advanced course work.
- pass a professional engineering exam (PE) for your given field; this test consists problems that test your ability to apply engineering knowledge to real world problems
Many states also have some continuing education requirements for license renewals.
Again, a PE is no guarantee that you know your stuff. But you are not allowed to sell your services directly to customers as an "engineer" without one. Working as an employee is a different matter. You really don't need one to help design that highway bridge, but someone in your company with a PE will have to vouch for the design when finished.
IMHO graduating engineers should strongly consider taking the EIT soon. It gets harder to remember all those fundamentals as years go by. You'll be better prepared if your career takes you in a direction where a PE is highly desirable. Having a PE also keeps more options open down the road.
All that said, the OP is certainly welcome to call himself a "network engineer". This isn't one of the recognized (licensable) engineering specialties. As others have quipped, there are no IT positions that have anything to do with traditional engineering. I'm NOT saying that the OP's job isn't difficult or challenging, but see why his friends tell him he's not a "real engineer".