I've always thought DNA's two strand were the same, because the ACGT are paired.
Let's take an example:
5' - . . . A C C G T . . . - 3' (+ strand)
3' - . . . T G G C A . . . - 5' (- strand)
Reading from the 5' direction to 3', as this is the direction in which RNA is transcribed, the two strands are distinct (...ACCGT... vs. ...ACGGT...). This results in different transcripts and potentially different genes, depending on the codons.
Just to make sure that there is no misunderstanding on the biology of what is happening because I think that is what the OP is asking. The DNA on the two strands are not the same. They are complementary and as such they will translate to entirely different proteins AAA translates to Lysine whereas its complement TTT will be translated to Phenylalanine
As zev points out so far we know that genes in one strand do not overlap with genes in another strand, having genes in both strands increases the number of genes an organism can have without having a disproportionally larger genome.
That said, there might be the case that two genes might perfectly overlap, there's not a physical constraint as long as both sequences carry a meaning in translation. It just hasn't been observed so far.
Thank you for all your answers.
I've thought that the genes came from the meaning strand, and the other strand didn't code any genes. It seems that the understanding of the two strands were not correct.
The process of transcription is somehow random, and each of the strands can be recognized as the template of genes. Is it right?