Isolated UART communication: Which side should send the supply for the isolator?
Since you seem to be running your UART on 3.3V levels from board 1 to board 2 we have to assume they are not far away from each other. That means it should not matter that much where you place your supplies.

There are other issues that will need your attention, namely:

-You actually need two power supplies for the Si86xx family. Refer to Si86XX CMOS DIGITAL ISOLATOR-BASED SERIALINTERFACE USER'S GUIDE.

-As Peter says in his comment above, there is a good chance you need an additional GND cable. You need a return path for the current running on TX and RX. With a non-isolated UART, oftentimes you would not need this return path because all equipment is connected to the same safety earth. But I'm sure you have experienced this problem yourself if you ever tried to make two handheld battery-powered devices talk to each other.

Just in case you want to take a look, I'm including a bunch of references I find useful:

-RS485 in practice

-TI Digital Isolator Design Guide

-Using DIGI products in the real world

-Analog Dialogue: Digital Isolators

-Serial Port Complete by Jan Axelson