It is possible to buy a car and use it without transferring ownership on the log book.
If you buy it from a dealer it will be already be registered, that is it will have number plates, etc. The dealer will also give you a document saying that he is transferring ownership to you. You should take these to KRA to transfer the log book. However it is perfectly possible to use the car without doing this, problems should only arise if you are required to prove ownership which is very unlikely in day to day usage. It might affect your ability to sell the car, though if you have the log book and the sellers document it may not, not, I don't know. There may be a very slight risk that if you have not transferred ownership someone may do so fraudulently to themselves, though presumably some one must be in possession of all the facts relating to the car to do this. (I have only heard anecdotal evidence of this, I have no personal knowledge).
If you buy a car privately, presumably the owner gives you the log book and may give something that says they are transferring ownership to you, (I have no personal experience of this, I bought both car and motorbike from dealers)
What you do have to do is take the car to an insurance company, have it valued and then obtain insurance coverage. It is the car that is insured, not the driver. You then display the insurance on the car's windscreen. If the police stop you they usually only check your driving licence and sometimes the insurance sticker, I've never known anybody be asked for the log book..

