Who is responsible for "blinding" a paper for double-blind review?
It will depend on the journal, and if the journal follows double-blind review, they will tell you how to properly format your paper. For example, the American Journal of Epidemiology:

Manuscripts must be submitted online in a double-spaced, blinded format, with pages numbered, in at least 12-point type, do not insert line numbers.
When uploading the manuscript to the Web site, please delete any text reference to the paper's authors, affiliations, and acknowledgments.