Powering up second LM386N results in screaming squeal
I don't know if anything else is wrong, but always bypass your power pins.  And I mean always, on everything.  In this case it means a 100nF cap between pins 6 and 4 of each chip, placed as close to the chip and with the shortest leads that are practical.

You should also have some bulk bypass -- I'd go with a 100uF electrolytic cap from +V to ground, just about anywhere on the board.