Most of the time the problem of depth is when people start legitimate digging for a drainpipe for instance.
And… then they find something. By coincedence. In that case you´re in for a lenghty yes/no game with the state for a settlement.
But what LaMa said, it is forbidden to dig deeper than “the plowline” without a good cause (urgent repair to said drainpipe) and/or permit.
You may keep the finds by the way, even if you report them they are not taken from you (a common misunderstanding).
Goes btw for everything you find. A wallet on the street, or whatever. As long as you report it. Used to be at a policestation, but that´s going to change now. You can report found/lost things online nowadays. (Yes, LaMa I follow teletext, papers and dutch tv here, so that latest bit of news of today is as fresh as it gets )
Report it, keep it and after a year it´s yours. A misconcepcion is that finders often think they have a legal right or something for a findersfee of 10%, unfortunatly, finderfee is at the discretion of the loser.
The State may offer to buy the treasure, but you are not obliged to sell it to them (they usually do a good offer though).
And most of the time, finds are not as dramatic as this one. So in most cases people are happy to sell it to the state or the museum that made an offer.
Other countries have other laws, of course.
Here on Menorca you are almost not even allowed to think of the word “metaldetector”. Let alone have one in your possecion. Doesn´t matter if you don´t use it, it´s simply forbidden. Every year some get confiscated. Parents often think that the little toy ones for their kids is ok, but it isn´t. And it makes for very bad start of your holiday.
And the fines are high.