Hi
yes, leave the 10K resistor, see what happens. It should work fine.
But... you might have another problem. Arduino 1 will not finish booting if it gets any serial data during boot from Arduino 2, and vice-versa.
The simple way around this is to use a TTL or CMOS gate with enable to turn the connection on once the Arduino's have finished booting.
Here's what's suggested in the FAQ:
Why doesn't my sketch start when I power up or reset the Arduino board?
Most likely because you are sending serial data to the board when it firsts turns on. During the first few seconds, the bootloader (a program pre-burned onto the chip on the board) listens for the computer to send it a new sketch to be uploaded to the board. After a few seconds without communication, the bootloader will time out and start the sketch that's already on the board. If you continue to send data to the bootloader, it will never time out and your sketch will never start. You'll either need to find a way to stop serial data from arriving for the first few seconds when the board powers (e.g. by enabling the chip that sends the data from within your setup() function) or burn your sketch onto the board with an external programmer, replacing the bootloader.
D