As individuals. 

The event originally celebrated being the political independence of the United States – plural, as it was in those much-forgotten days – from Great Britain. But we now live in the United States – singular – in which a consolidated federal authority has, over time, all-but-eliminated the former independence of those states, who were united, once upon a time, in the rhetorical rather than political sense. As in, united to gain their individual independence from Great Britain.

They were not – in those days – a united political entity, except loosely.

Jefferson and Adams, for example, routinely referred conversationally to their respective countries – i.e., to the states in which they lived. They would not have comprehended a consolidated United States, singular.

That took another four score and seven years to "correct"  . . . by force of arms.

People will debate the inevitability of the foregoing; the flaws – perhaps traps, deliberately set – within the Constitution that replaced the Articles of Confederation, the latter clearly referencing something other than a consolidated singularity. That is for historians to weigh and consider. Perhaps the whole object of the struggle for independence was not for independence – of the individual, from government – but rather just another political tussle between factions for control of the government.