I've found that Russian-made DVDs of "classic Soviet films" often do have subtitles both in English and in Russian -- for example, I own Вий, Иван Василевич Меняет Профессию, and Ирония Судьбы on DVD, and they've all got Russian subtitles.

But with more recent Russian movies, it seems, the DVD producers often neglect to provide Russian subtitles on DVDs, even though it would be extremely simple to do so. From Googling, I found this Russian-language site для глухих и слабослышащих ("for the deaf and hard-of-hearing"), which complains:

Но поскольку русских субтитров на DVD-дисках очень мало (особенно в отечественных фильмах), то глухим приходится скачивать субтитры из Интернета. Вот тут и начинаются проблемы. Многие плееры отображают наложенные субтитры безобразно.
"But since there are very few DVD disks with Russian subtitles (especially for domestic [Russian] films), deaf people have to download subtitles from the Internet. And that's where [technical] problems begin. Many DVD players display added-on subtitles improperly."

So it's a known problem (that article is from 2011), but for whatever reason, the Russian movie industry doesn't seem in a hurry to provide subtitles for Russians with hearing difficulties, who have to rely on "kludgy" solutions from the Internet that don't always work well.