A B737-300 Flight Crew failed to properly document the recurrence of a chronic #2 Reverser Unlock warning. In addition they failed to write up the failure of the #1 reverser to deploy to the next flight crew. The latter failure the result of Maintenance in-oping both reversers rather than just the chronic #2.

Narrative

The #2 Reverser Unlocked light illuminated in flight. The reverser had been MEL'd the day prior for the same problem; but had been cleared overnight. The light extinguished during descent and the thrust reverser operated normally upon landing. Contract Maintenance could not find anything wrong. I discussed the situation with Dispatch and Maintenance; and since the next leg was to a city with Company Maintenance; we decided to continue on and let Maintenance look at it there. En route to our destination; the light illuminated again; extinguished on descent; and the thrust reverser operated normally when we landed.Maintenance locked out the #2 thrust reverser and entered the MEL in the logbook. We then flew to our next destination. Upon landing; I raised the #1 thrust reverser handle to reverse idle; but could not get the thrust reverser to deploy. I briefed the outgoing Captain and explained what had happened. He said something to the effect of; 'since they were going to a city where the thrust reverser wouldn't be used in calculating stopping margin; he would check it out on the next leg'. I agreed. I failed to write up the #1 thrust reverser for not deploying. The next day I found out that Maintenance had in fact disabled both thrust reversers; not just the #2 thrust reverser.Whenever there is a problem with the aircraft; the Captain needs to investigate thoroughly and find out what's going on. After landing; I should have called Maintenance right away and not given the airplane to the next crew until we had determined why the #1 thrust reverser did not work properly.