Suppose in group 1 there are $n$ people and in group 2 there are $n$ people. If we conduct a 2 sample t test versus a paired t test, would we use $2n$ for power calculations in the two sample t test and $n$ as the sample size for the paired t test?
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In general, the way that input is specified depends somewhat on the equations or software that you use to calculate the statistical power. But yes, it would be normal to specify the sample size in paired-samples t-test as the sample size $n$, and for independent groups t-test it would be the sum of the two group sample sizes, which when equal would be $2n$.
If you take post-hoc power analysis in G-Power 3 for example, you specify the sample size for paired-samples and you specify the individual group sample sizes for independent groups t-test.
The big additional factor for power analysis in paired samples t-test is the correlation between the two levels of the repeated measures factor. Conditional on the null hypothesis being false, the size of the correlation is positively related to the amount of statistical power.