Does the circuit value problem require only log space in alternating Turing machines
It is a classical result that ALOGTIME=P. You can find a proof in lecture notes of Madhu Sudan. Both the circuit value problem and the monotone circuit value problem are P-complete, and in particular in P, and so can be solved in alternating logarithmic space.

One thing which might be confusing you is that we only measure space on work tapes. Space on the input tape is for free, though this comes at a price: you are not allowed to modify it.