Imputing missing genotypes from separate genotyping panels
Given that you mention wanting to use 1000 Genomes as a reference panel for imputing genotypes into your two SNP chip panels, I am going to assume that you are working with human data. 

In that case there are several options you can go with:


If your two panels are of European descent, then you are probably best off using the HRC reference panel together with a fast genotype imputation tool such as Beagle 4.1 to impute genotypes in each of your two SNP chip panels separately.
If your panels are not of European descent, then you will likely want to use the 1000 Genomes phase 3 reference panel with Beagle 4.1, Impute2, or Minimac3.


In either case, there are two phasing services available that will do much of the heavy lifting for you 1,2.

The second Wellcome Trust Case-Control Consortium paper performed a cross-imputation analysis as you describe.  I don't see many studies using multiple SNP chip panels.  You will need to take care in your analysis that you are not hit by batch effects from using two different SNP chip panels.

Also, none of these methods will work if the region you are imputing into has too few variants.  I'm not sure what the minimum number of variants is, but if you are using a whole genome genotyping panel of at least 500k SNPs, then you should be ok if you impute a whole chromosome at a time.