Kidd played most of his junior career with the Brandon Wheat Kings where he became one of the top amateur prospects in the world. His best year was 1989-90 when he was named to the WHL East First All-Star Team and chosen the CHL Goaltender-of-the-Year. That summer he was the first round pick, 11th overall, by the Calgary Flames. Toward the end of the 1990-91 season he was traded to the Spokane Chiefs where he helped the team win a Memorial Cup.
Kidd also excelled internationally during his amateur days. He was a member of Team Canada at three World Junior Championships. In 1990 he backed up Stephane Fiset when the team won gold and was the starter when they repeated as champions the next season. He also played well in 1992 but the team lost many starters from the previous tournament and fell to a disappointing sixth place finish.
The Flames loaned Kidd to the Canadian national team in 1991-92 and he served as the team's back-up to Sean Burke at the 1992 Winter Olympics. He played in a 10-0 rout in Norway as Canada went on to win the silver medal. Kidd also split a pair of decisions with the Flames that season. In 1992-93 he adjusted to the pro game as a member of the IHL's Salt Lake Golden Eagles.
Kidd joined the NHL on a full-time basis by playing 31 games for Calgary in 1993-94. As a rookie he posted a 3.16 goals against average and 13 wins. Following the owners' lockout in 1994-95, Kidd led all NHL goaltenders by playing in 43 games but the team was eliminated in the first round of the playoffs. He played 47 games in 1995-96 but Calgary was now a team in decline. He recorded three shutouts and a 2.78 goals against average but could only muster a 15-21-8 won-lost-tied mark.
On August 25, 1997 he and veteran forward Gary Roberts were sent to the Carolina Hurricanes for Andrew Cassels and netminding prospect Jean-Sebastien Giguere. Once again Kidd's numbers were good with three shutouts and a 2.17 goals-against average but the team was average. During his second year with the Canes, Kidd was reduced to second string status behind Arturs Irbe.
In June 1999 Kidd was claimed by the expansion Atlanta Thrashers then traded to the Florida Panthers, another decent but not dominant team. In 1999-00, he split the goaltending chores with veteran Kirk McLean. The next year Kidd shared the goaltending work with young phenom Roberto Luongo.
With the emergence of Roberto Luongo, Kidd's playing time was reduced to 33 games and he subsequently signed as a free agent with the Toronto Maple Leafs in the summer of 2002. He backstopped the Leafs to a 6-10-2 in his 2002-03 season and returend the following year to post a 6-5-2 record. His short stint in Toronto came to an end when during the NHL lockout he played with HC Orebro 90 in Sweden. The following season Kidd signed with the Hannover Scorpions in Germany.