In my still relatively short journey into the world of electronics, I have stumbled on multiple schematics that use resistors in serial communication and suspect that they are there for some kind of protection but I have never been exactly sure.
Regards your schematic, I think the reason might be is to avoid two data transmission chips/systems clashing or fighting. It appears that if the external TX pin is active then it will win the battle to talk to the chip on the right i.e. the chip on the left is defeated.
Other uses of resistors in series with data transmission output pins are usually to provide impedance matching and to prevent reflections - they are usually in the range 10 ohms to 33 ohms.
Depending impedance of cables and risk of contention on Tx ( Shoot thru across supply rails), overshoot can be minimized when the source and transmission line and terminations are matched. That said, I recall ARM ( 74ALCxx) Zouts are 25 Ω +/-50% over temp and shielded twisted pairs can be 50~100Ω single ended typ. and often 120Ω differential. Matching depends on PCB layout with controlled Z.
I would use 25 Ω on each source with 50 Ω traces and 100 Ω load as a tradeoff to signal swing and overshoot but your choice of signal integrity may differ.
They are probably trying to protect against a dead short on the exposed header. If you look at the internals of the chips they are very small resistances, by putting those resistors in line you don't affect "normal" operation, but protect the outputs from being over-driven.
Some embedded systems, required to pass "susceptibility testing" and thus exposed during testing to high RF field energy, will get UPSET at certain test frequencies. I've diagnosed a number of these systems, brought to me by other engineers wondering what to do, and my (successful) approach is to find the long traces and insert 10Kohm resistors, at one end or the other, next to the IC that is being UPSET.
Some of these systems had chopped-up GND systems (not even planes) and the gaps between PCB fill regions let the regions act as two halves of a dipole antenna. Any trace crossing over became vulnerable. The resistors rescue the project. [edit: We discuss separate GND regions as dipole antennas, examining how to ADD short traces as jumpers across long traces that caused the GND to be slit. In other words, if you MUST cut the GND with some long trace, rejoin the GND pieces together in several spots using short jumpers above the evil trace.]