How to use angular-ui-bootstrap (modals) in typescript?

The instance injected by angular will be of type ng.ui.bootstrap.IModalService.

And since you are using "controller as" syntax, there is no need to start using $scope here, instead you can use resolve as shown in the angular-ui modal example.

Here's the example code:

class ItemsListController {
    static controllerId = 'ItemsListController';
    static $inject = ['$modal'];

    data = [
        { value1: 'Item1Value1', value2: 'Item1Value2' },
        { value1: 'Item2Value1', value2: 'Item2Value2' }
    ];

    constructor(private $modal: ng.ui.bootstrap.IModalService) { }

    edit(item) {
        var options: ng.ui.bootstrap.IModalSettings = {
            templateUrl: 'modal.html',
            controller: ItemDetailController.controllerId + ' as modal',
            resolve: {
                item: () => item // <- this will pass the same instance 
                                 // of the item displayed in the table to the modal
            }
        };

        this.$modal.open(options).result
            .then(updatedItem => this.save(updatedItem));
            //.then(_ => this.save(item)); // <- this works the same way as the line above
    }

    save(item) {
        console.log('saving', item);
    }
}

class ItemDetailController {
    static controllerId = 'ItemDetailController';
    static $inject = ['$modalInstance', 'item'];

    constructor(private $modalInstance: ng.ui.bootstrap.IModalServiceInstance, public item) { }

    save() {
        this.$modalInstance.close(this.item); // passing this item back 
                                              // is not necessary since it 
                                              // is the same instance of the 
                                              // item sent to the modal
    }

    cancel() {
        this.$modalInstance.dismiss('cancel');
    }
}

Am I right to assume that both ItemsListController and ItemDetailModalController need an instance of the same scope in order to exchange the data?

Yes. I actually think of modals as an extension of ItemsListController containing members shownDetails:ItemDetailModalController. Which means you don't need to come up with a messy way of sharing $scope as its just a single $scope.

And how can I define the controller and the template for the modal directive using the interfaces above?

This is what I have (note that you are sharing the scope):

            this.$modal.open({
                templateUrl: 'path/To/ItemModalDetails.html',
                scope: this.$scope,
            })

Where $modal:IModalService corresponds to what angular strap gives you : http://angular-ui.github.io/bootstrap/#/modal