First of all it’s nice of you to have reported it – or even want to report it. Most would have just taken it as a home decoration. This is what makes a human differ. Gongradulations.
In case you are still here and you have your findings:
1. The museum has an archeologist and he is really good at what he does. He could for sure tell you if it is something of a value. His office is right below the entrance yard, in a small white-stoney house, you can see it as you park outside the museum.
2. Kalymnos is full of small pottery-pieces scattered around the island. Greater example is Kastelli, where there has never been a reconstruction project of the fortress, therefore everything is still there. Pottery, stones, etc. Since you found it in Dodoni, probably it’s not ancient but could easily go back in the late Byzantine – knights’ era. As you can easily see there is an old settlement there, which used to be vivid until the early 20th century. So, this could be from 500 to no older than 100 years old.
3. In general, just because Greece (and Kalymnos) is full of antiquities and history could be found everywhere – anywhere, don’t be surprised if they tell you that it’s of no value! I think only if something (such as small pieces of broken pottery) is from the classical era is kept.
Anyway, congadultions once more and please try to find mr. Michalis Koutellas (archeologist in charge). He will guide you through. Museum phone number: 22430 23113