It's possible to interrogate people before they're reported missing with consistent success, but the mechanics aren't clear at all, and you can't just muddle through it and achieve an ideal result.
You basically need to get a psychologist with high int/heart/psychology and low wisdom, then train him up in a few ranks of each cultural skill at the university by taking physics/economics/religion courses. Once he's got a few ranks of everything, have some expendable liberals with cheap guns drop pick up lines on unimportant conservatives like secretaries and janitors. When you're making your dating decision, look for low wisdom/psychology/culturals and kidnap them if they're all low. Have a bunch of liberals tend to the captive in addition to the psychologist, and have the psychologist repeatedly attempt to convert using props (and nothing else). If you've racked up enough skill, use plenty of guards and props, and selected a suitably weak conservative, you should be able to turn him over about four days, and psychology skill trains really quickly during interrogations, so you'll be able to work your way up without a lot of fuss.
If you have to resort to drugs or beatings, you are either aiming too high or aren't adequately prepared. You haven't got a hope of converting the conservative before his absence is reported, so just abandon the safehouse and figure out what you're doing wrong. Don't try to convert people if you don't have a decent lead interrogator. Don't kidnap random conservatives; always check their stats through dating. Don't beat your prisoners.