Film Societies tend to get their films in two ways.
They contact a film distributor and ask for permission to screen a film, using either a DVD that they own or a DVD that the distributor will rent out,
If a film society can demonstrate to a distributor that it is not-for-profit, subscription based and doesn’t charge for admission, it is eligible to access that distributor’s DVD concessional rate. All of the distributors listed on the site offer discounted rates. This includes Amalgamated, Roadshow and Madman and many others.
They borrow a DVD or 16mm film from The Non-theatrical Lending collection (NTLC) of the National Film and Sound Archive (NFSA). Anything in the Collection can be screened without you doing any copyright clearence because the NFSA has done that for you. It is also much much cheaper.
However, the NFSA will hold a film society or club liable for any damage that occurs to the 16mm film. It is best to have film handling insurance, which comes free with membership of the Federation.