Emitter Follower will give an output impedance much lower than the ESR of the LED to give optimal response.
If Rs =50 Ω driving base with Beta=100, Re=0.5 Ω to Anode and Cathode to ground. Caution to not overdrive LED with proper offset to Vf and swing of 0.5V (e.g. Red Led Voffset=1.25V with swing from 1.2 to 1.7 will give a faster square wave response to at least 10MHz. Adjust offset by 0.1 increments to match diode.
Add current limiting resistor in collector to protect LED.
All the voltage swing in the LED due to the internal ESR , whereas the Vf threshold is typically measured at the 10% of rated current or 2mA in this case.
It is possible to get at least 2 orders of magnitude of light out and the LED and maybe 3, but reception is limited range.
For optical communication a narrow bandwidth such as 16QAM or 64QAM is better for the receiver such as used by ADSL modems and dialup modems.
It is better to have the directivity of a laser diode or the higher current capability found in IR diodes.
Daylight blocking filters reject ambient light so IR is preferred.
If you like to experiment, make a fibre optic link using mono-filiment heavy fishing line and drill hole between two LEDs with 1/32 drill and use clear epoxy . Connect a TI transimpedance amp with emitter follower and a switch for both ends. YOu can use the LED as a Photo Diode but performance is much less than typical 0.5mA/mW of a good PD. Now you can create a half duplex fibre optic link.