The entire radium decay chain* emits only alpha and beta particles (and no gamma radiators)
Alpha particles are blocked by a piece of paper, so these are mainly dangerous if you get the material inside your body (if you for example lick a brush with radium paint on it) and beta radiation is slightly better at penetrating materials but is generally stopped by a very thin sheet of metal (like the metal body of a watch).
There is also very little radium in a watch (very thin layer of paint that contains radium and a material that absorbs radiation and then emits that energy as a photon, ie visible light).
*Radium collapses into radon->Polonium 218->Lead 214->Bismuth 214 etc etc. Each time it emits either a helium-4 particle (alpha radiation) or an electron/positron (beta radiation).
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