UPDATED VERSION FOR REST API v2: https://gist.github.com/Shelob9/e9c8ca48a37ba4c39796

jp-api-post-editor.js

(function($){
    var url = JP_POST_EDITOR.root;
    url += '/posts';

    function getPostsByUser( defaultID ) {
        url += '?filter[author]=';
        url += JP_POST_EDITOR.userID;
        url += '&filter[posts_per_page]=20';

        $.ajax({
            type:"GET",
            url: url,
            dataType : 'json',
            success: function(response) {
                var posts = {};

                $.each(response, function(i, val) {
                    $( "#posts" ).append(new Option( val.title, val.ID ) );
                });

                if ( undefined != defaultID ) {
                    $('[name=posts]').val( defaultID )
                }
            }
        });
    }

    $( document ).ready( function()  {
        getPostsByUser();
    });


    $( '#select-post' ).on( 'submit', function(e) {

        e.preventDefault();
        var ID = $( '#posts' ).val();
        var postURL = JP_POST_EDITOR.root;
        postURL += '/posts/';
        postURL += ID;

        $.ajax({
            type:"GET",
            url: postURL,
            dataType : 'json',
            success: function(post) {

                var title = post.title;
                var content = post.content;
                var postID = postID;

                $( '#editor #title').val( title );
                $( '#editor #content').val( content );
                $( '#select-post #posts').val( postID );
            }
        });


    });

    function results( val ) {
            $( "#results").empty();
            $( "#results" ).append( '<div class="post-title">' + val.title + '</div>'  );
            $( "#results" ).append( '<div class="post-content">' + val.content + '</div>'  );

    }



    $( '#editor' ).on( 'submit', function(e) {
        e.preventDefault();

        var title =  $( '#title' ).val();
        var content =  $( '#content' ).val();
        console.log( content );
        var JSONObj = {
            "title"         :title,
            "content_raw"   :content,
            "status"        :'publish'
        };

        var data = JSON.stringify(JSONObj);

        var postID = $( '#post-id').val();
        if ( undefined !== postID ) {
            url += '/';
            url += postID;
        }

        $.ajax({
            type:"POST",
            url: url,
            dataType : 'json',
            data: data,
            beforeSend : function( xhr ) {
                xhr.setRequestHeader( 'X-WP-Nonce', JP_POST_EDITOR.nonce );
            },

            success: function(response) {
                alert( JP_POST_EDITOR.successMessage );
                getPostsByUser( response.ID );
                results( response );
            },
            failure: function( response ) {
                alert( JP_POST_EDITOR.failureMessage );
            }
        });

    });

})(jQuery);

jp-api-post-editor.php

<?php
/*
Plugin Name: JP REST API Post Editor
*/

add_shortcode( 'JP-POST-EDITOR', 'jp_rest_post_editor_form');
function jp_rest_post_editor_form( ) {
	$form = '
		<form id="select-post">
			<select id="posts" name="posts">

			</select>
			<input type="submit" value="Choose Post To Edit" id="choose-post">
		</form>

		<form id="editor">
			<input type="text" name="title" id="title" value="Hello there">
			<textarea id="content" ></textarea>
			<input type="hidden" name="post-id" id="post-id" value="">
			<input type="submit" value="Submit" id="submit">
		</form>
		<div id="results">

		</div>
		
	';
	if (  is_user_logged_in() ) {
		if ( user_can( get_current_user_id(), 'edit_posts' ) ) {
			return $form;
		}
		else {
			return __( 'You do not have permissions to edit posts.', 'jp-rest-post-editor' );
		}
	}
	else {
		return sprintf( '<a href="%1s" title="Login">%2s</a>', wp_login_url( get_permalink( get_queried_object_id() ) ), __( 'You must be logged in to edit posts, please click here to log in.', 'jp-rest-post-editor') );
	}

}

add_action( 'wp_enqueue_scripts', 'jp_rest_api_scripts' );
function jp_rest_api_scripts() {
	wp_enqueue_script( 'jp-api-post-editor', plugins_url( 'jp-api-post-editor.js', __FILE__ ), array( 'jquery' ), false, true );
	wp_localize_script( 'jp-api-post-editor', 'JP_POST_EDITOR', array(
		'root' => esc_url_raw( get_json_url() ),
		'nonce' => wp_create_nonce( 'wp_json' ),
		'successMessage' => __( 'Post Created Successfully.', 'jp-rest-post-editor' ),
		'failureMessage' => __( 'An error occurred.', 'jp-rest-post-editor' ),
		'userID'    => get_current_user_id(),
	) );

}