I want to produce silhouette I need to calculate 10^8 distances
So this question is rather about (resources for) computation a big square distance matrix than about Silhouette criterion. Yes you need all the distances to be able to compute silhouette values: this criterion is matrix-based.
Is silhouette really practically used?
Yes, sure. Cluster analysis - for your information - is used not only for big data.
But if you are doing specifically K-means clustering on big data then silhouette is not the best choice. K-means is all about centroids and variance and therefore criterions such as Calinski-Harabasz are more focused here.
I'm using K-medoids variant, so I'm not sure if that would work? Also,
I'm using a specific distance measure (Dynamic Time Warping)
Standard (original, by Kaufman & Rousseeuw) Silhouette Statistic is based on average distance between a point of interest and points of a cluster (its own cluster or an alien cluster). Averaged distances is perhaps the most "universal" or "neutral" (assumption-free) measure of cluster closeness or within-cluster density. If so, then you may use Silhouette to assess partitions produced by any clustering method (provided that you can - your data is not too big).
On the other hand, the Silhouette formula itself is general and so the criterion could be made to reflect other than average-distance notion. For example, my SPSS macro (found on my web-page) computing silhouette implements also nearest-neighbour, farthest neighbour, and distance-to-centroid notions. Thus, Silhouette criterion can exist in a number of versions; some version is better suited for one clustering method, another version - for another clustering method.
It is possible to program a special distance-to-medoid and ultimately distance-to-whatever-your-fancy versions of silhouette criterion. All one will need is to supply all the distances to those "quasi-centres" of clusters. And by the way, then - if you get all the distances from each point to each "centre" - you won't need at all the matrix of pairwise distances between all the points, and a bigger data could be happily processed.
The formula of Silhouette itself and the most common (average-distance) version of the index is not tethered to a specific distance measure. However, bred specific other versions of it could appear tethered.