What you are asking about is gene expression, not the physical pairing of chromosomes. And the answer is...it depends.
Your body is able to mark certain chromosomes for use and others for disuse. For instance, one of each X chromosome in each cell of a female is made into an inactive form called a Barr Body. In each cell, however, a different X may be inactivated.
In autosomes, either chromosome may be used. For instance, individuals may have AB blood because both the A allele on one chromosome and the B on the other are being expressed. In some cases, in a given cell, one chromosome's allele may be more likely to be expressed than another because of modification of the DNA after replication.
There is also evidence that in some cases, your body may deliberately target certain members of autosome pairs to be inactivated, but that is still being researched for a clearer understanding.