I assumed we were talking about private ownership vs. National Guard (or "regulated" State Militias). The Founders did mean that individual citizens should be armed.
"To preserve liberty, it is essential that the whole body of the people always possess arms, and be taught alike, especially when young, how to use them."
- Richard Henry Lee, Federal Farmer No. 18, January 25, 1788
"The Constitution shall never be construed to prevent the people of the United States who are peaceable citizens from keeping their own arms."
- Samuel Adams, Massachusetts Ratifying Convention, 1788