Usually yes, if you live in the state where the LLC is registered and have a physical address there. If you live somewhere else, you generally cannot, and you need an agent located in that state.
The rules come from each state's statute. Florida's is typical: section 605.0113 says the registered agent must be an individual who resides in Florida and whose business address is the same as the registered office, or an entity authorized to act as one. Delaware goes a step further and lets the LLC itself be its own agent under section 18-104, but still requires a registered office in Delaware. Neither works for an owner living in another state or another country.
The part people underestimate is availability. Florida section 48.091 requires the registered office to be kept open from at least 10 a.m. to 12 noon and 2 p.m. to 4 p.m. every weekday except legal holidays, with someone there who can be served. If you are your own agent, that someone is you, at that address, during those hours, for as long as the company exists.
Missing a delivery does not stop a lawsuit. Under Florida section 48.062, if the registered agent cannot be served after one good faith attempt, process can be served on any member or manager of the LLC, or on anyone listed on its latest annual report. The case moves forward either way. The only thing that changes is whether it reaches you in time to respond.
It is also a matter of public record. Florida's statute requires the state to keep a record of each registered agent and registered office and to hand that information out on request. If the registered office is your home, your home address is public, and it shows up in business databases that copy state records.
So: if you live in the state, work from a fixed address with regular hours, and do not mind that address being public, being your own agent is legal and costs nothing. If you live out of state, travel, work from home, or want service of process to reach someone whose only job is to pass it on, a commercial registered agent is what that yearly fee pays for.