Ancestry is a privately owned site, so no. The Mormon church currently has an agreement to do family history information sharing with Ancestry, and for this reason is able to obtain free Ancestry memberships for church members. But IIRC, the agreement is set to expire at some point.
When you figure many returned missionaries and current temple workers, BIC TBM fill positions at Ancestry dot com, it wouldn't surprise me in the least there is overlap between church work and genealogy in the private sector. Such as turning in names for the dead, etc.