This is how I do this:
1. Make that custom post type translatable
a) WPML –> Translation Management –> Multilingual Content Setup –> Custom posts
b) Check “Translate” for those custom post types you want to translate. For example: “Places”.
c) “Save”
2. Enable slug translation for the site
a) WPML –> Translation Management –> Multilingual Content Setup –> Custom posts slug translation options
b) Check “Translate custom posts slugs (via WPML String Translation).”
c) “Save”
3. Select to use different slugs in different languages
a) WPML –> Translation Management –> Multilingual Content Setup –> Custom posts
b) For your preferred custom post type (p.e. “Places”) you can check “Use different slugs in different languages for Places.”
c) Read the issue with this option on my first missage on this thread.