The closest private competitor to SpaceX right now IMO is Rocket Lab. They are the only company that has an orbital rocket that has a path forward for reusing the first stage right now. Rocket Lab's rockets are already pretty cheap for launching small sats and when they start reusing their first stages they might be able to get their cost per Kg to orbit close to what SpaceX can offer right now. But Rocket Lab's rocket is tiny compared to the Falcon 9, even SpaceX's first rocket, the Falcon 1 could put more payload into orbit than Rocket Lab's Electron rocket. It's not clear if Rocket Lab has any plans for making a bigger rocket.
Blue Origin is the only other company (besides the Chinese) that is currently building reusable first stages. Blue Origin's BE-4 engine also looks really promising and was designed from the beginning to be reusable. I think a lot of people are giving up on Blue Origin because they have yet to launch anything into orbit and they've been around longer than SpaceX. If Blue Origin does get it's New Glenn rocket off the ground and can recover and reuse the first stage they have a good chance of being cost competitive with the Falcon 9/Falcon Heavy.
The reality though is if SpaceX is successful with their Starship/Superheavy project they will be so far ahead in the cost/reusability department it will take a competitor a decade or more to catch up. Keep in mind part of what makes Starship even possible is the Raptor engine which SpaceX started developing in 2009. It's likely SpaceX has spent hundreds of millions of dollars just developing the Raptor, the Air Force chipped in over $40 million to the Raptor program at one point and SpaceX has built over 30 Raptors already at a cost of about $1 million a piece.