We can measure both the number of particles emitted and their energy
Radioactive elements emit several types of radiation. The major types are alpha particles (2 protons and two neutrons or, in other words, a helium nucleus), beta particles (which are electrons) and gamma radiation (basically high energy X-ray photons). These have different abilities to do harm regardless of their energy (so even paper will block most alpha particles but you might need a large lump of lead to stop gamma radiation).
The first thing that is easy to measure is the amount of radiation. Sensitive detectors can measure every individual radioactive emission passing through them. So we can measure the count of emission events. More radioactive elements have a higher emission count. All other things being equal, a lower half-life equates to a higher emission count. The SI unit for this is the becquerel which is one emission per second.
Measuring the danger from radiation is much harder than just measuring how much of it there is as different types of radiation do different amounts of damage to people or equipment. Alpha particles are much nastier than the same number of beta particles, for example. So the units for this are more complicated and there are more of them. The simplest is the gray (which is 1J of energy absorbed per kg of stuff). But the damage done depends on more than just the energy so there is another unit for biological exposure that accounts for that called the sievert.
The actual energy per emission can be measured by using a variety of instruments. Specific radioactive decays have characteristic emission energies which are often measured in electron volts (eV) which are an energy unit suitable for describing the energy of very small things (an eV is about 1.6×10-19<\sup> joules). Gamma photons have an energy of about 40 keV, beta particles ~500 keV and alpha particles ~5,000 keV.
How dangerous a particular isotope is depends on a combination of things: the energy of the particles it emits; how easy it is to stop that particle before it hits you; and how much damage that particle will do to you if you absorb it.