Sounds like pretty good advice all around.
I agree being shards and likely shards are plentiful, I doubt there would be any repercussions from keeping them, but I am not familiar with the local laws.
Museums and universities are where you are most likely to find someone who really knows their stuff about this. I'm not an archeologist, and even if I were I don't think I could identify the shards with any certainty without holding them in my hands. Certainly the Mediterranean is loaded with pottery, but if that particular site is exceptionally plentiful it may indicate more than a random find of a shattered pot. Something to keep in mind is that these "ancient" looking pots are also made in fairly recent times, as souvenirs and decorations and in some cases an attempt to deliberately fake expensive pieces.
If the pottery market is anything like the coin market, old Roman and Greek coins are so plentiful, that even their ancient age really doesn't give them much in the way of value, and the relative ease of faking them makes them quite easy to purchase on the secondary market inexpensively. But there is always that unscrupulous dealer looking to hook a novice.
If it were me, and I liked the pieces for their decorative value or sentimental value, I would probably just set them aside as a souvenir to remember my trip to that place, and just consider their "value" secondary. If such finds were quite rare in that general area, or if there were legal complications in keeping them, then I would either leave them there or notify the local archeological authorities and leave the rest to them.
I hope you get to enjoy them. Great find!