Both Widget Logic and Dynamic Widgets are plugins that allows controlling where widgets display. You can use it to specify which post types, pages, categories or tags widgets are allowed, including which languages they will display in.
You can evaluate these plugins under Plugins > Add New by searching their names, or directly here:
Dynamic Widgets
For example, you may be using a widget-driven home layout and need the HTML content of text widgets or widget descriptions to be translated, not just the titles and category selection which is inherently possible using the String Translation extension. Either of these plugins allow you to setup one copy of each plugin in each language, then specify the language to link them to.
For a detailed walkthrough on using Widget Logic, see:
http://wpml.org/2011/03/howto-display-different-widgets-per-language/
To use Dynamic Widgets, the same process applies. To select the language, you would click the “Static” link in the widget once it is setup, which will bring you to the Dynamic Widget settings page for that widget. Expand the Languages section and select the language to display the widget in, then save.
An alternative tool for translating websites and widgets is Weglot, find out more about Weglot here.