E-commerce platforms support the needs and livelihoods of their two most important stakeholders -- customers and producers/sellers. Multiple algorithmic systems, like ``search'' systems mediate the interactions between these stakeholders by connecting customers to producers with relevant items. Search results include (i) private label (PL) products that are manufactured/sold by the platform itself, as well as (ii) third-party products on advertised / sponsored and organic positions. In this paper, we systematically quantify the extent of PL product promotion on e-commerce search results for the two largest e-commerce platforms operating in India -- Amazon.in and Flipkart. By analyzing snapshots of search results across the two platforms, we discover high PL promotion on the initial result pages (~ 15% PLs are advertised on the first SERP of Amazon). Both platforms use different strategies to promote their PL products, such as placing more PLs on the advertised positions -- while Amazon places them on the first, middle, and last rows of the search results, Flipkart places them on the first two positions and the (entire) last column of the search results. We discover that these product placement strategies of both platforms conform with existing user attention strategies proposed in the literature. Finally, to supplement the findings from the collected data, we conduct a survey among 68 participants on Amazon Mechanical Turk. The click pattern from our survey shows that users strongly prefer to click on products placed at positions that correspond to the PL products on the search results of Amazon, but not so strongly on Flipkart. The click-through rate follows previously proposed theoretically grounded user attention distribution patterns in a two-dimensional layout.