Yes, actually, it does very clearly mean individuals. To quote the relevant amendments that cite "The People":
9th:
The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.
10th:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
What meaning would the 4th have, if "The People" were state assemblies? "The right of the people to be secure in their persons"... Would that mean it was up to individual states to decide if you are secure in... What, exactly?
Or consider the 1st, "the right of the people peaceably to assemble, and to petition the Government for a redress of grievances"... Why would a state need to petition "The Government", as it is "The Government".
There is no reasonable reading of the constitution that reads the second amendment as anything but an individual right.
Or for instance, does it mean that the federal government can't ban state organized militias?