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You have three options:

1. Register a dynamic sidebar for a specific page, and then create a page template that calls dynamic_sidebar() for the registered sidebar.
2. Use an is_page( $id ) conditional wrapper in your page.php template
3. Output Widget code conditionally, using is_page( $id )

Custom Page Template

The first option would include your register_sidebar() call in functions.php:

register_sidebar( array(
    'name' => 'Specific-Page Sidebar',
    'id' => 'specific-page-sidebar'
    // other array parameters omitted for brevity
) );

Then, you create a custom page template, e.g. template-test-sidebar.php.

Then, in the appropriate place in the custom page template, invoke the specific sidebar, via dynamic_sidebar():

dynamic_sidebar( 'specific-page-sidebar' );

is_page() conditional in page.php

The second option is more simple, but less dynamic, as it requires you to hard-code the page ID into the template. Simply wrap dynamic_sidebar() in an is_page() conditional:

if ( is_page( $id ) ) {
    dynamic_sidebar( 'specific-page-template' );
}

Conditional Widget Output

The third method involves conditional output of the Widget itself, which generally requires use of a Plugin, such as Widget Logic or any of several other related Plugins.

With this method, you don’t need to register a separate dynamic sidebar. You simply use one of your existing sidebars, and then the Plugin provides an in-Widget option to specify the conditional logic to use, such as is_page( $id ) to determine if the Widget should be displayed.