Hi Dario,
STAR does not use strand information in any specific way. In principle, I could have used it to filter out spliced reads with "wrong-strand" intron motifs (for example CT/AC intron for a read that maps to the "+" strand). However, all strand specific protocols produce some amount of wrong-strand calls. With the classic dUTP-based protocol (which I believe is also the basis of the Illumina stranded TruSeq) we typically get 0.5%-1% mis-stranded reads. I believe it's better to map the reads in a un-biased fashion, and - whenever possible - filter out the "antisense shadows" later on.
The mates in the Aligned.out.sam are assigned strands (i.e. 0x10 bit of the FLAG) according to their sequences, this is the standard SAM convention which is not very convenient for RNA-seq. Concordantly mapped mates have opposite strands. In the classic dUTP protocol the strand of the 2nd mate coincides with the original RNA strand, while the strand of the 1st mate is reverted.
For visualization of BAM files, you may want to revert the strand of one of the mates (1st mate for the classic dUTP protocol), so that UCSC browser colors both mates the same way.
If you are using bedTools genomeCoverage tools, you would need to revert the strand of the 1st mate.
Cheers
Alex