Joel Sherman of the New York Post reported the union has pushed for a right to privacy among its players, leaving it to them to disclose whether they are out of the lineup due to contracting the virus. Several players have tested positive for COVID-19 since teams began reconvening for next month's beginning of the 2020 season but few player names have been made public.
It's unclear how MLB will go about not revealing a coronavirus-related reason for placement on the IL. MLB teams regularly disclose the reasons that players go on the injured list, along with their prognosis and expected length of absence. If a player is absent for multiple weeks without a designation, MLB runs the risk of outing a player's status without even disclosing it publicly.
A possible solution could be MLB stopping the disclosure of why players are on the injured list altogether. However, that would be something that would need to be collectively bargained. Failure to disclose injury-related absences would have implications with league partners and media.
A player does not have to test positive to go on the COVID-19 list. Known exposure to the disease and/or a medical recommendation could land the player on the list.