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Yes, but that would not be very good strategy for either team. The most common reason for pulling a goalie would be if your team is down by a goal or two with just a few minutes to play. By pulling the goalie the team would get an extra skater. If both teams pulled their goalies it would be 6 on 6 hockey with no goalies. That would just mean players would be shooting shots from their zone or center ice. Again, you can pull two goalies at the same time, but it would be meaningless.
From the 2009-2010 NHL Official Rules:
81.4 Line Change on Icing - A team that is in violation of this rule shall not
be permitted to make any player substitutions prior to the ensuing
face-off. Should the offending team elect to utilize their team time-out
at this stoppage of play, they are still not permitted to make any player
substitutions. However, a team shall be permitted to make a player
substitution to replace a goalkeeper who had been substituted for an
extra attacker, to replace an injured player, or when a penalty has
been assessed which affects the on-ice strength of either team. The
determination of players on ice will be made when the puck leaves the
offending player's stick.
Yes, the goalie may return to his/her net when the player replacing him/her returns to their bench.