I'm a second year student and have completed 3 clinical placements so far. I'm not sure if you are supposed to actually hand it in to someone, but rather keep the card with you when you are on clinical placement. The aim is to have a record of things you are vaccinated against in case there is some accident at work.
So far, no one has ever asked to see my vaccination record card thing, but i strongly advise you to have it completed.
What I did was go to my GP and have a blood test done (as someone mentioned earlier) to determine the what I am and am not vaccinated for. After that, if you're missing out anything, the doc can give you the shots and off you go.