Right-back Dumbuya passed a medical with the Magpies on Tuesday morning and will travel with his new team-mates for their pre-season tour of Finland.
The 25-year-old moved to the UK at a young age and caught the attention of Doncaster in 2009 before joining Portsmouth last summer.
Dumbuya spent the first half of the campaign at Fratton Park before linking up with Crawley in January on a six-month deal.
Magpies boss Chris Kiwomya said: "Mustapha is a player I wanted to bring to Meadow Lane. I've seen him play, I've spoken to him and he has the right attitude, he wants to play for this club and he wants to do well here.
"That's key for me, I need players here who want to be here, who want to succeed for this club, work for their team-mates and for the fans who pay their money to come and watch them every week.
"Mustapha will add an extra dimension to our play, he is quick, he can play in a number of positions, he can be a threat down the wing but also a good defender, he should fit in well to the style of play that I will be looking to play this season."
Meanwhile, left-back Sheehan has put pen to paper on fresh terms which will keep him at Notts County next season.
The 26-year-old saw his two-year contract expire at the end of June and both he and the club were keen to strike a new deal.
Sheehan said: "I'm delighted to be here and we've got an exciting year coming up, where hopefully we can kick on from last season. I wanted to be here and I'm looking forward to working for Chris, he's got some big ideas and it's hopefully going to be a good place to be. He wants to sign a good few players and we've got a good base here to be pushing high up the table."