The Ulster skills coach played in the same Tigers team as Cockerill two decades ago, and their respective teams face off in the Pro14 on Friday.
Malone told the BBC: “I went to his wedding so we must have had quite a good relationship by the end of it, but he used to bully me to pieces.
“He really scared me - frightened me, actually,” Malone added.
Edinburgh are due to face Ulster twice in as many months, and currently sit three points behind Jono Gibbes’ side in Conference B, and Malone is wary of the improvements Cockerill has made to the capital outfit.
Admitting he was “amazed” when the former England international succeedeed Alan Solomons as Edinburgh head coach, Malone added: “I couldn’t really see the link between him and Edinburgh when it was announced, but he’s done really well.
“You can see his characteristics coming through. [In the past], Edinburgh had good players, but they didn’t seem to sort of gel or fight it out the way they do this year.”
Likening the current Edinburgh side - which was today boosted by the news that three players have put pen to paper on new deals - to an “old-fashioned Munster team”, Malone declared them the “best Edinburgh team we’ve played in a few years.”
He continued: “They look tough and rugged and they didn’t always look like that.
“That’s what characterises [Cockerill’s] personality and I think he’s had an instant impact, making them more aggressive and more competitive.”