Yes, every LLC formed in Wyoming (or any state) must have a registered agent with a physical address in that state. This is a statutory requirement under Wyoming LLC Act § 17-28-101 et seq. The registered agent receives legal service of process and official state correspondence on behalf of your LLC.
It does not defeat the privacy purpose. The registered agent’s address appears on your public filings, not your personal address. That is the whole point. You hire a registered agent service (typically $50–$150 per year) and their address is what shows up on the Wyoming Secretary of State records.
Some registered agent services also offer nominee organizer and nominee member/manager services. This means their name appears as the organizer on the articles of organization, adding another layer of separation between the LLC and your personal identity.
One important caveat: the Corporate Ownership Transparency Act (COIA) reporting under FinCEN now requires most LLCs to file a Beneficial Ownership Information (BOI) report. This is not public, but it is reported to the federal government. So while Wyoming gives you privacy from public searches, the government still knows who you are.