When we refer to the "Rosette API", we're talking about a typical SaaS cloud-based web API. The Rosette API supports the majority of functions of the SDK -- the API is a subset of the SDK in most regards. The major features the SDK has that the API lacks are stateful and user-customized features - mainly Name Indexes (to disambiguate against a custom curated list of names, rather than the simple pairwise matching the API is limited to) and model training -- our entity extractor can be trained against a dataset/model of your own in the SDK, but not in the API. However, the API contains Sentiment and Categorization, which are not fully featured SDKs just yet.
A newly-available product, really a "third flavor" of our text analysis suite, is "Rosette API On-Premise". Users can basically run their own web server with all the same functions as the Rosette API, but handle it internally (which solves cloud security worries), and customize (models/indexes) as needed.
Rosette can be easily customized to fit into the system you need on-premise -- please ask us if you have any questions. We have an experienced team ready to help integrate Rosette into your system.