The exact details may be a matter of national law, but in general, you are required to have proof that you can support yourself to enter Schengen countries (this, in fact, is true of many countries in the world). This requirement is spelled out in the Schengen Borders Code:
For stays not exceeding 90 days in any 180-day period, a non EU-country national must:
justify the purpose of his/her intended stay and have sufficient means of subsistence;
This applies whether or not a visa is required for entry. There is no requirement, and indeed no mechanism, to submit this information in advance.
For a specific example, see the Government of Switzerland, which requires of non-EU/EEA/Swiss Citizens:
They have sufficient financial resources to cover the entire duration of their intended stay as well as the return trip to their country of origin, the transit trip through a third country, or can obtain the necessary financial resources by some other lawful means.
As a practical matter, the vast majority of individuals will pass passport control with few or no questions. Those singled out for additional scrutiny, based on their profile, demeanor, answers to preliminary questions, and whatever other criteria the border guards apply, may be asked for significantly more information, including proof of funds.
This need not be cash in hand; debit and credit cards are acceptable. In the unlikely event that you are asked for such proof, it's a good idea to have some means of providing it, which could be as simple as showing your bank balance on your bank's website or app or by carrying the appropriate documents.
A few Schengen countries even require that visa-exempt travelers have adequate health insurance for entry too.