NOTE! This answer from Ashutosh Pandey is now misleading. At least for anyone using samtools >= 1.0 (which was released ~6 months after the answer was posted).

Since 1.0 (released Aug 15, 2014), we have the (somewhat confusingly named) --ignore-overlaps option:

-x, --ignore-overlaps  Disable read-pair overlap detection.

With this option, samtools mpileup 1.x behaves like it did in 0.19, but the default behavior is to NOT DOUBLE COUNT overlapping paired end reads.** It has been this way for the past 3.5 years.

You can try it yourself:

$ samtools
Program: samtools (Tools for alignments in the SAM format)
Version: 1.0 (using htslib 1.0)

$ cat test1.sam
@HD     VN:1.3  SO:coordinate
@SQ     SN:chr1 LN:248956422
seqid   163     chr1    39415   0       9M      =       39418   12      ATGTGTTTT       AAEE6EEEE
seqid   83      chr1    39418   0       9M      =       39415   -12     TGTTTTGCA       E<EAEEA<<

$ samtools mpileup test1.sam
[mpileup] 1 samples in 1 input files
<mpileup> Set max per-file depth to 8000
chr1    39415   N       1       ^!A     A
chr1    39416   N       1       T       A
chr1    39417   N       1       G       E
chr1    39418   N       1       T       i
chr1    39419   N       1       G       Q
chr1    39420   N       1       T       i
chr1    39421   N       1       T       e
chr1    39422   N       1       T       i
chr1    39423   N       1       T$      i
chr1    39424   N       1       g       A
chr1    39425   N       1       c       <
chr1    39426   N       1       a$      <

$ samtools mpileup -x test2.bam
[mpileup] 1 samples in 1 input files
<mpileup> Set max per-file depth to 8000
chr1    39415   N       1       ^!A     A
chr1    39416   N       1       T       A
chr1    39417   N       1       G       E
chr1    39418   N       2       T^!t    EE
chr1    39419   N       2       Gg      6<
chr1    39420   N       2       Tt      EE
chr1    39421   N       2       Tt      EA
chr1    39422   N       2       Tt      EE
chr1    39423   N       2       T$t     EE
chr1    39424   N       1       g       A
chr1    39425   N       1       c       <
chr1    39426   N       1       a$      <

Note that for this to work right, the paired read names must match (which they do for Illumina data).

The overlapping bases in the paired end reads will be counted for both reads. In the above example, they will be counted twice.