You can have separate TX and RX antennas and not run a duplexer, but there's some issues that will make a duplexer look like a really attractive option.
To run without a duplexer, you need a LOT of vertical separation between the TX and RX antenna, not a few feet, but tens of feet. That might make your system coverage very lopsided unless you have access to a tall tower. The lower antenna isn't going to work as well. Then you'll have a bunch of coaxial cable and its losses to deal with.
You can do horizontal separation, but the distances will be greater. Again, more cable to deal with. To overcome feed line losses, you'll need to spend a lot of money on good cable, like 1/2" or larger Heliax, and that can be a few bucks a foot. Connectors are expensive, too.
On the other hand, finding a used duplexer on line and getting a knowledgeable person to tune it for you will save you some money. You'll only need one antenna and one feed line.
If the antennas are too close, the transmitter will desensitize the receiver and your coverage will positively suck.