The amount of current that a source can draw is going to depend upon the power-supply internal resistance, the wiring and connectors resistance, the switches and circuit traces all leading to the ESP8266 module and the internal resistivity of the IC bonding. Substitute "battery internal resistance" for power-supply if your project is battery operated.
With lab powersources, I have monitored ESP8266 pulsed current as high as 850mA !!! With the RF section active (receive) and programmed by Arduino, the idle current is about 80mA. If one disables the RF section and just uses the ESP8266 for Arduino at 80MHZ, then the current is about 40mA.
Far too many breadboards and jumper-pin cables cannot be used to prototype on the ESP8266 because the board compression connectors are too resistive and the inexpensive pin-wires are cheap and have high resistance. Prototyping this way will definitely cause you to go crazy as inexplicable things will happen!
If you breadboard, use high quality breadboards such as those by Global Specialties (example only.) Do not use pin-jumper connectors .... use 22 gauge solid copper wire, freshly stripped. Before inserting an ESP8266 module, use a quality Ohm meter to check all power connections from the source power unit to the breadboard - resistances should be in the low sub-1 Ohm range.
If you use batteries to power your project, use batteries with a known low-Z construction: never use carbon-zinc. Quality alkalines may work, but LiOn or LiFePO4 are excellent choices. If you must use a low-quality product for supplying input power, use a quality DC-DC "buck" converter just before the ESP8266 to bring a 5V - 24V input into the 3.3 Volt range for the ESP. A quality "boost" converter can be used with voltages lower than 3.3V to provide a low-Z stable power to the ESP8266. Do not screw up your design by using quality batteries, DC-DC units, and then use cheap slide switches and cheap thin hook-up wire.
Ray