For an ESP8266 application with a decent quality 5V source, you can get by with just a single 47-1000 μF cap on the output side, just place it physically as close to the ESP power pin as possible.
But what you can get by with is, of course, not the same as what is recommended. Capacitors serve the important functions of smoothing out voltage fluctuations and also filtering out errant signals (high/low frequencies), both of which can cause erroneous operation of a microcontroller and the voltage regulator itself.
For this reason, ideally both the input and output sides of your AMS1117 linear voltage regulator will have 2 capacitors in parallel -- one with a high capacitance (1 to 100 μF) for filtering low-frequency noise and one with a low capacitance (0.01 to 0.1 μF) for filtering out high-frequency noise.
Extra benefit can be achieved by using the proper capacitor chemistry for each application -- ceramic for filtering high-frequency and tantalum for filtering low-frequency. Standard can-style polarized electrolytic capacitors are not ideal. This is why AMS1117 modules like the one OP is asking about exist; they provide the best possible characteristics for the widest range of applications without any of the guesswork.
See: Arduino Forum discussion of AMS1117 application with ESP8266
See also: this StackExchange question