Melbourne, thanks for offering to help people. This may help you:
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Common sources of support for people in general as well as persons traumatically exiting a religious situation in particular are government-sponsored Social Services and in some communities various churches. So you may begin by contacting those two sources and ask what kinds of assistance they already offer, if any, that would be helpful for exiting persons, and write down what they say as well as the names of their representatives, their phone numbers and emails for easy reference.
After discovering what they already offer you will know better what kind of other support is needed. In general, though, someone who has been kicked out of their home over religious differences may be in need of not only some kind words of hope and encouragement, but also some short term material support.
For example a kicked-out JW (or Mennonite etc) may need a toothbrush, walked through how to set up and keep their own bank account, temporary shelter if not in a home then at least by being allowed to park and live in a car by your home, etc.
The emotional and spiritual components should also not be ignored.
Insofar as the emotional goes, it may be better to not tell them to talk about what happened so much as to offer examples from your own experience or that of others, then "if" they want to reveal things they will feel freer to do so.
Some people do not want to talk about it and for them that may be the best way to cope, whether you agree and like it or not. Those who do talk may not simply talk but vent anger and frustrations. Decide in advance if you yourself are emotionally strong enough to hear it.
If not, then gently let them know upfront that you are there to help them, but unfortunately not able to bear hearing a lot of negativity that may cause yourself discomfort or flashbacks. Be prepared to suggest either a trained a counselor or, after you have done this for a while, someone whom you know from experience has been able to hear such venting in the past.
Spiritually it is normally not wise to try to convert an exited person to your own beliefs or you will probably be resented as a parasite trying to prey on them in a time of depression. Instead emphasize that you wish to assist regardless of what they do or do not believe.
You may mention (if such applies) that you still read the Bible and are a Christian, or they may be able to easily figure that out on their own from seeing publications etc which you may have. Also on their own they may or may not ask about your beliefs, which if any church you attend or would recommend and the like.
Depending on the individual and their situation, they may or may not immediately or later appreciate an offer to sit and enjoy reading or hearing a few scriptures with you in a non- or trans-denominational manner. But, again, above all, don't be pushy but rather be available.
As a Support Group person, you are offering to act as a friend, not a Church Group converter. If however you feel you must emphasize the need to convert exited persons to your own religious beliefs, or in fact are a Preacher, Minister, Priest or other clergy person, then you need to make that gently but clearly known from the beginning to be honest to both yourself and those whom you seek to help.
You will need to advertise the fact that you exist. On bulletin boards, in brief newspaper ads etc you can give a further contact telephone number and or email. The key (note carefully) is to advertise regularly.
If you speak with a person over the telephone you can tell much about them so as to determine how best to help them.
For example, an agitated person who has just exited may contact you while high on drugs or alchohol, and you may see that it would not be best to then tell them they can come stay in your own home for a few days where they might pose a threat to yourself or other family members. Having the needed names, phone numbers and addresses ready, you can get such a person help from a social service or at least postpone speaking and meeting them in-person until later when they are more sober. On the other hand, the person may and indeed probably will be very upset but clearly pose no danger and you might be able to immediately and personally do more for them.
Your sometimes having to refer people to professional does not diminish your own role as a person who actually was in the same religious group. Indeed your insights may prove crucial, even life-saving for you may be able to get through to and positively help those who feel severely crushed and depressed in such a way that someone who has a PhD in counseling but was a never a member could not. Still, know when to involve a professional.
Even if never formally trained as a counselor then by attending free or inexpensive training sessions or talking to the experts and reading whatever they may offer, you may be able to learn a lot, for example, how to recognize if a person has suffered a mental break-down, is clinically depressed, etc. Indeed, with time you might even decide to become certified as a professional.
Regardless, try to be realistic and flexible and also become progressively knowledgeable so as to help more and more persons who are survivors of abusive religion.
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Melbourne, I am part of the internet Free Christians movement which offers a free newsletter (Free Christians News) on the Bible, relgion, science, health etc and for those wanting it also new spiritual fellowship based on LOVE rather than forced sameness of beliefs.