It's important to distinguish SIRT1, which is a gene, from the protein it encodes, sirtuin 1 (some papers use SIRT1 for the protein too, but to avoid confusion, I'll use Sirt1). When looking at compounds that affect gene expression, one method of measuring the effects is measuring the change in mRNA levels for that gene when the compound is administered, usually with northern blotting or qPCR. In this case, because the gene encodes a protein, quantifying the protein levels also tells us about the expression of the gene.
The SIRT1 gene doesn't have any regulatory role itself, so in this case, people are more interested in the Sirt1 protein. While we can measure protein concentrations, Sirt1 is an enzyme that deacetylates proteins and resveratrol has been proposed to directly alter the activity of the enzyme (it's also known to increase SIRT1 expression, but this isn't the controversial aspect), so it's important to examine the effects of resveratrol on Sirt1's ability to deacetylate certain proteins. The way this is done is to expose various proteins/peptides to Sirt1 and look at whether they deacetylate. One method of measuring this is to use a special peptide as the substrate to which a fluorophore can be attached if the acetyl group is cleaved.
The reason resveratrol's effects have been debated is that the early studies that showed large activation effects were later found to only occur because of the fluorophore on the substrate and not resveratrol alone. Some work since then using other detection methods have shown activation effects: This paper tried many different substrates on a microarray using fluorescently labelled antibodies as well as kinetic measurements by mass spectrometry and found that resveratrol increased Sirt1's activity for some substrates, decreased it for others, and left many the same, though they didn't offer much in the way of structural insights that might be responsible. I haven't done a thorough literature search, but it seems like the debate is less contentious now that people are using other assays to examine activity.
Just to summarize your questions, it's not hard to quantify whether a molecule increases the expression of SIRT1, but because Sirt1 is an enzyme, any kind of label you add to measure activity could have its own effects on that activity. Any detection method that can measure the deacetylation of substrates could potentially be used for studying Sirt1 activity, and there are many, but label-free techniques like mass spectrometry are especially useful because they reduce complications in simulating biological conditions.