Shamans shapeshifted through trance only, which means that they were astral projecting.
They would, however, dress up as the animal and learn to mimick its movements to understand how it would be, to be that animal. They would wear the skin or other parts of it (teeth, claws, ets) and act as if they were the animal to the best of their ability. In doing this, they would let themselves go, and become engulfed in the energy/essence of that animal. Those who saw them, at times believed they saw the shaman shapeshift. Imagine seeing a bear's entire skin stand up and walk off like a man after swaying and grunting like a bear. You have to understand, those who saw "shapeshifters" were outsiders who did not understand the culture they were observing.
The cultures didn't write down their knowledge, so their religion and practices could not be interpreted unless the language barrier was broken and great trust was earned. This was rarely ever accomplished. Most of the information we have today has been handed down through oral tradition from generation to generation, until finally, someone recorded it down for public eyes to see.
When a medicine man could connect extremely well with the behavior and thoughts of an animal by pretending to be it perfectly, he was believed to be able to become one with the animal in the spirit world and communicate to the animal in the physical.