We estimate theoretically the strength of DNA-DNA electrostatic friction forces emerging upon a slow drag of one DNA over another one in a close juxtaposition. For ideally helical DNA duplexes, this friction occurs due to correlations in electrostatic potential near the DNA surface. The latter originate from the intrinsic helicity of DNA phosphates and adsorbed cations on a scale of 3.4 nm. They produce positive-negative charge interlocking along the DNA-DNA contact. For realistic nonideally helical DNAs, where electrostatic potential barriers become decorrelated due to accumulation of mismatches in DNA structure, DNA-DNA frictional forces are strongly impeded. We discuss possibilities of probing the DNA-DNA intermolecular interactions in strongly confined DNA superhelical plies, as obtained in single-molecule experiments.