POINT I/O can get a little confusing if you just look at the detail diagrams for the I/O modules. Take a bit of a step back and look at the various ways that the POINT I/O Selection Guide 1734-SG001 illustrates power distribution. It does a good job of explaining the different ways the EP24DC, the EPAC, and the FPD modules distribute and separate power.
When you connect 24V DC to the 1734-AENT's terminal block, that power goes both to the internal electronics and the communication POINTBus, as well as to the Field Power bus along the POINT I/O assembly.
There is a DC/DC converter that provides 5V to the POINTBus and the AENT adapter itself. That's the 1 Ampere value that's been discussed.
You can see the Field Power bus on the 1734 bus connector; it's the two larger slider connectors closest to the terminal blocks.
Depending on the type of module, +24VDC and DC Common are brought through the POINT I/O modules to the terminal blocks. Some modules route both +24V and DC Common to the terminals, some route just one, and some route neither. This was the hardest thing for me to wrap my head around; I was looking at the power bus and assuming that the terminal blocks somehow connected directly into it.
In the case of the 1734-OB4E, you get four terminals for the Outputs themselves, and four DC Common terminals you can connect to the common side of each load.
You could connect a hefty 10 A DC power supply to the POINT adapter and run everything along the POINT power bus, or you could use separate power supplies for the network adapter and the bulk power, using a 1734-FPD to isolate the AENT adapter from the power bus.