I queried Azuga (maker of one of the approved devices) and received the following response:
Thank you for writing in and for your interest in the OReGO Road Usage Charge program, we really appreciate it!
Hope you are doing well!
The OReGO Road Usage Charge (RUC) program is a potential gas tax replacement to fund our roads and bridges. It is a stable and long-term funding/revenue collection method to maintain roads and infrastructure in the state of Oregon.
Our device is extremely light in its use of the vehicle's battery (the device's power draw). This has been very carefully tested over many years in many applications. We have one of the lowest, if not the lowest, standby voltage draws in the industry. Furthermore, the device itself detects the voltage of the vehicle battery. If it senses an issue or a weak battery it goes into an even deeper standby mode to ensure it is not affecting the battery.
Post ignition off and broadcasting the trip end message, the device goes into a sleep mode with minimal device voltage draw. While the engine is on, the device is sampling GPS and cellular connectivity, therefore, it consumes more power. When the device broadcasts the message to our server while the ignition is ON it uses relatively more power to transmit data. But of course, the battery is being charged while the engine is on.
There are some vehicle makes and models that have minor electrical/wiring issues and, in a way, "throw off" power on their own when they detect things like OBD2 port devices or other automotive 12V accessories. It is possible this could affect the battery, but these are extremely rare cases. However, if you leave a device in a vehicle without driving for over a month you may see some additional power loss from your battery. Most vehicles, though, would have already drained the battery due to internal electrical components in that time frame. We recommend that any vehicle that is seldom used either be driven every two weeks or hooked up to a smart battery charger every two weeks to replenish the lost capacity.