transformerless ethernet
The transformer may be necessary for some drivers to feed the supply voltage, but it could be replaced by two small chokes in this case.
Generally in 10BASE-T (10 MBit Ethernet), the transformer is also a low pass filter, not absolutely required but necessary to get the specified waveform from square wave at the driver. At the receiver, a low-pass filter increases noise immunity.
In 100 MBit Ethernet, the transformer is really a transformer, and it's 100ohms 1:1 matched impedance usually. For that reason it could be placed anywhere in the differential line between connector and PHY and for isolation purposes one would be enough instead of two. For the said on-board connection it could be omitted and replaced by capacitive coupling at each receiver, to my opinion. Bothside 100 Ohms termination should be used as with a transformer. If you are able to control the transmitter PHY, you could decrease transmitter current and modify the matching to reduce power consumption, however.
You should also consider, that the receiver has considerable gain reserve to compensate for line losses.