The first sentence makes sense. I would just use 'putting' instead of 'putting out'. You could also say 'check every name matches with the correct number.'
The second sentence also makes sense, but it is a little bit long because you have 'name' repeated 3 times, and 'on the' repeated twice. For the second one, a better sentence would be 'When putting sheets on the desk, check that each name matches the one on the name tag'.
The last sentence also makes sense, but there is definitely a better way to say it. 'The thing is' at the beginning is usually used when you are giving an opinion about something. It is also very informal and never really written. For that sentence I would write: 'the name on each desk should match the one on each sheet you hand out, so you mustn't mix them up.'
I hope this helps!