To expand a bit, generally speaking:
If you are are in Australia
and keeping the boat
in Australia and assuming it has an engine
of more than 6HP then it should be registered in a state. To buy the boat
get a receipt from the owner and have them fill in the relevant States transfer of registration form, pay any relevant fees
and you are done. If it's on a trailer
you will need to transfer that rego too.
If it is an Australian Registered vessel (ie. Federal Registration through AMSA) and you are Australian citizens then get the owner to fill in the AMSA transfer forms. If you keep it in an Australian state you will still need to register it in the relevant state. If you are not Australian then get the owner to cancel the Australian registration, there is a form for that too from AMSA. All this info is available on the AMSA website.
If you are Australian and plan to take the boat to another country then you will need to register the boat with AMSA, if not already done so.
If you are not Australian then you may want to register the boat in another country, if you plan to take it overseas. Alternatively if you plan to leave it in Australia and you have a residence address in the relevant state then just a State registration will do. Nationality is not relevant for State registration. As an example here is the link to the Queensland
State transfer form, read the first page carefully.
In terms of a private sale, if the boat has a State registration number you can search the registration
You can also check the vessels State registration status - for example
You can also check the PPSR for any mortgage on the vessel https://www.ppsr.gov.au/searching/do-boat-search