If the soldier didn't read it himself, then it was read to him. So, yes, the average soldier either read it or heard it read.

The Declaration of Independence was ratified on July 4, 1776, in Philadelphia. It was taken to a local printer that night who printed 200 copies, as broadsides, which were then distributed to various people around the country.

One of those people was General George Washington, who was stationed on Manhattan (New York City) with the Continental Army. At the same time, General William Howe was anchored with the British Royal Navy across New York Bay on Staten Island.

The Declaration of Independence was read to the Continental Army on the evening of July 9, 1776. That broadside delivered to Washington and read to the troops is now in the collection of the Library of Congress.

Three days earlier, on July 6, 1776, the first newspaper published the Declaration in full, the Philadelphia Evening Post. On August 2nd, it was printed in the South Carolina and American General Gazette down in Charleston, South Carolina. By then, the Declaration had been printed in complete form in all the major cities in the Thirteen Colonies. If you scroll down on this article, it gives a longer list of all known newspapers' first printings of the Declaration, and Harvard University did a "Declaration Resources Project" that actually mapped the Declaration's spread graphically.

So, the active duty Continental Army mostly heard the whole thing on the night of July 9, 1776, by which time George Washington had read it himself.

Any soldiers who joined the Continental Army later, or any soldiers who fought in their local county militia, very likely either read it or heard it read when their local newspapers published it in full, which had been done in most major cities within a month of its ratification. It may have taken a little longer to hear about it if you lived far in the countryside, but it was big news, so most people would have taken the time to read or listen to the whole thing. At the very least, they would have heard all about independence and the gist of the reasons given in the Declaration.

EDIT: Fixed link.