Since you’re connecting Serial RX of the first device to TX of the other, and TX of the first to the RX of the other, either device should be able to talk to and AND listen to each other. I hope you have kept in mind that the ESP8266 is a 3v3 logic-level part, and that the Arduino Uno is a 5v logic-level part, so you need to use a level shifter of some description - where it be a voltage divider or a dedicated level shifter breakout/IC.
And really, since this is a crossover, they should actually be able to talk at the same time - it is just a matter of structuring the code so it can handle that - you don’t don’t the code to block - just keep sucking in the message from the other side until it hits the \n or whatever ‘end of message’ delimiter you use.
The other problem you might face is that of debugging - since if you use the hardware serial port on each device, how do you do any debugging of what is happening with the device, without interfering with the two devices talking together. The solution is to use software serial on both devices, leaving the hardware serial free on both for debugging messages. This post from the Arduino forum might help… serial communication between esp8266 and arduino - Programming Questions - Arduino Forum