What do we mean when we say a transistor amplifies current?
For an NPN BJT in forward active mode, the voltage drop across the base-emitter junction is typically 0.6 - 0.7 V.

The voltage drop from collector to emitter is whatever it takes to pull the expected collector current from the rest of the circuit. (That means, to calculate the drop you treat the collector-emitter branch as a current source and calculate what voltage that produces at the conductor due to the rest of the circuit; then check to be sure all voltages and currents are consistent with your assumption of forward active operation)

If the c-e voltage gets too low, then the gain (whether you want to quantify it as a current gain \$\beta\$ or a transconductance \$g_m\$) falls until at about 0.2 V from collector to emitter you end up in saturation operating mode rather than forward active.