In a civil case, the proper name of the defendant need not be known and it can simply be filed against "the individual with this phone number" and their identity possibly later subpoenaed from the phone company. But what about in a criminal prosecution, private or public?
Those wishing to instigate a private prosecution must follow Part 7 of the Criminal Procedure Rules 2020 beginning with Rule 7.1(1)(a), when...
a prosecutor wants the court to issue a summons or warrant under section 1 of the Magistrates’ Courts Act 1980...
The relevant requirements in Section 1 of the Act are that..
... the justice [i.e. magistrate] may issue—
(a) a summons directed to that person requiring him to appear before a magistrates' court to answer the information, or
(b) a warrant to arrest that person and bring him before a magistrates' court.
It follows that if the prosecutor cannot, or will not, identify that person then the court cannot issue a summons or warrant (as the case may be) against that person.
The name of the accused must appear in the indictment or information. See R. v. Turmel, 2005 QCCA 6, interpreting Criminal Code, s. 581:
Nowhere is it mentioned that the name of the accused must be included in each count: the fact that the name of the accused appears in the heading of the indictment is sufficient.
Practically, this is one of the barriers to private prosecutions. Private individuals do not have the same investigatory resources and powers as government to allow them to identify people that they will charge.