Mountain Rescue is not always completely free on the continent (though in France, it currently is). I am covered for personal injury and professional liability but you are strongly recommended to obtain your own insurance before heading out. You will need to be covered for trekking activities in Europe.
For UK residents, the British Mountaineering Council (BMC) has excellent mountain holiday insurance policies, for example.
For residents in France, it is best to be insured via the Club Alpin Français
For residents in Spain, we recommend insurance via the Federación Española de Montañismo
We also remind you that it necessary to bring your European Health Card (ask your GP if you don't have one). This is currently free of charge and entitles you to the same health-care rights as local residents when travelling through Europe. However, be careful as (contrary to popular belief) health-care is not 100% free in all EU countries. In France, for example, the state only covers you for 60% to 70% of costs and even local residents will have to take out their own private insurance to cover this. You may also want to make sure this is included in any additional travel insurance you take out.
I remind you that « climbing, hill walking and mountaineering are activities with a danger of personal injury or death. Participants in these activities should be aware of and accept these risks and be responsible for their own actions and involvement » (BMC participation statement).