I'm measuring about 1A in bursts of about 1ms in my application right now. A little guesswork is required to make the measurements, so the results might be off by say up to 50%, but the order of magnitude is significantly different from the 300mA that is normally quoted.
Although I can't see exactly what happens, I think this happens when a packet is transmitted.
The image currently on my scope has a 800us linear droop of the 3V line, followed by a "recovery" in almost 200 microseconds. A probe on the 0.4ohm current-shunt shows that about 800mA of current is being consumed during both the droop and the recovery period. (and that current stops immediately as the recovery completes).
I think my MCP1802 is in current-limit mode during the whole droop-recovery period. At about 800mA, while the datasheet mentions something like 380mA typical. Weird.
From the fact that the recovery is about 4 times faster than the droop, 800mA is about 4 times more than the "deficit" that occurs during the droop. So the total current during the droop is about 0.8 + 0.2 = 1.0A.
My question: Can anybody confirm that the ESP will draw about 1A from the 3.3V line in bursts when transmitting?