The duo played under Kevin Lynch at the non-league side prior to Lynch recently taking up his coaching role at the Proact Stadium.
Spireites boss Paul Cook told said: "We're having a look at them and they're having a look at us and that's how it works.
"I don't like the word trialists because it's like the lads are under pressure, isn't it?
"Then if they don't get taken on, they've failed. But there's plenty of reasons why lads don't get taken on that isn't always to do with their ability. So I like to say they're training with us."
Brown was a junior at Tranmere before moving to Manchester City where he became reserve-team captain. He has also played for Northwich Victoria.
Shaw came through the ranks at Marine and was their player of the year during the 2011/12 campaign.