logic gates with 2n222 transistors and leds not working
simulate this circuit &ndash; Schematic created using CircuitLab

I have annotated your schematic with the nominal voltages expected across LEDs and transistors and the resulting currents. 
However these are the values I start with, to estimate what I expect to happen in our circuit.

I have started from the top, subtracting out the fixed voltages (VLED, VBE, VCE) until I get to a current setting resistor R1, then I can estimate the current.

Red LEDs have nominal 1.8V across them, transistor VBEs 0.6V, VCE_sat 0.3V. 
Actual values would be different as the currents are so low- try the simulator feature.

Your transistors are emitter followers, and have gain of, lets say, 100, so the base current of the transistors will only be 1/100th of the emitter current 106/100 - 1uA

So I am impressed you can see D1,2 light at all. 

Keep eating the carrots, they are obviously working



Obviously you can't have the D1,2 in series with the bases.

This arrangement is still flawed for an AND gate, as the base current of Q2 goes to the output, regardless of the state of Q1.
It would works as an OR gate (Q1,Q2 E and C connected), but not an AND gate.

As a NAND gate, you would put the load in the collector.



Have a look at some datasheets for 74 series logic (7400) as the old datasheets for TTL,CMOS (and RTL) have the internal schematics.