This is kind of a complicated question in the broadest sense, so I won’t post very long on this — because I’m ill-equipped to give an answer. New York, for example, has a cold-weather shelter guarantee, as do many other cities (Boston doesn’t send anyone away in extreme cold weather conditions either). Generally speaking, if you Google your city’s name and ‘homeless services’ or ‘homeless cold weather,’ you will likely find some type of resource — a van, a shelter policy, or a phone number. You’ll also probably find this post on City-Data referencing how bad it is to be homeless in Chicago, Minneapolis, Detroit et al and why Arizona and California tend to have “more drifter people” (I’m not sure that’s scientifically true, although there is a semi-famous story of Mayor Bloomberg sending NYC’s homeless to Florida during the winter).
When the amount of beds becomes an issue, typically there’s some kind of money to be found for temporary lodging in extreme cold — you see this right now in Vancouver, for example. In places like Anchorage and even the Bay Area, though, deaths do mount. One issue is simply societal: a good portion of homeless people may have reached a point where they legitimately distrust the main authority aspects of society. In that case, if a cop or a van encourages them to go inside, they may believe they have better odds on the streets. There’s also the aspect of alcohol and drugs (in some cases), and the aspect that, in larger cities, it’s impossible to patrol every area where a homeless person could be seeking shelter, especially at night. Some of these issues are addressed in Forbes, which also quotes Harold Pollack at UChicago encouraging you to “play the long game” in terms of contributing to the right organizations in your area rather than worrying about specific homeless people on the streets one night. (For more on Pollack’s observations about the poor, read this.)
National Health Care for the Homeless Council has some resources, but really your best bet is to look into local options — what does your city provide? What do churches provide? Where are shelters? What are the options? (Your best broader bet, like Pollack says, is to invest in these issues rather than ignoring them.)