WASHINGTON—Congressional Democrats on Saturday scrambled to assess the scope of an unprecedented leak that revealed the personal cellphone numbers and some email addresses of nearly 200 current and former Democrats in the House of Representatives.

Following the posting online of a spreadsheet containing the personal information of lawmakers and senior staff members late Friday, House Minority Leader Nancy Pelosi (D., Calif.) notified all House Democrats in a Saturday email that they were invited to participate in a 6 p.m....