The gym is privatizing its membership, opening only to invited clients.

privatizing 是动词 privatize 的现在分词形式。privatize 由形容词 private（私人的）加上动词后缀 -ize 构成，意思是"使私有化、将...转为私人所有或经营"。在句中表示"正在将会员资格转为私人专属"的过程。