How does one typically purchase annuity products?

Annuity products are created or "manufactured" by insurance companies.  That said, annuities are almost always distributed through intermediaries who are financial advisors of some sort.  In other words, you would purchase an annuity from a licensed financial advisor.

At a minimum, annuity sales require proper licensing.  Fixed annuity sales require an insurance license.  Variable annuity sales require both insurance and securities licenses.  There is legislation pending that would make securities licensing mandatory for the sale of equity indexed annuities.

There are hundreds of thousands of financial advisors and intermediaries.  Financial advisors may work independently as registered investment advisors (RIA), they may work at a bank or a credit union, they may be independent or captive insurance agents, they may work as stockbrokers at firms such as Morgan Stanley, or they may work at regional investment firms. 

Not all financial advisors and products are created equal.  It makes sense to shop around for both advisors and product recommendations.  Try to find and advisor who is patient, consultative, and wants to discuss the financial planning process before making any product recommendations or pitches.