How can I check if the current user is accessing his/her own user page?
There are two paths a user could be visiting their own page: "user" and "user/UID" where their UID matches.

There is a global variable $user that stores the currently logged in user.

So in the variable preprocessor for the template you can determine whether or not a user is on their own user page and save it to a variable that you can reference in the template file, like below.
Note the use of $variables['logged_in'], which is an available variable as listed in the page.tpl.php documentation.

/**
 * Override or insert variables into the page templates.
 *
 * @param $variables
 *   An array of variables to pass to the theme template.
 */
function THEME_NAME_preprocess_page(&amp;$variables) {
  // If we are on the 'user' page or a sub-page of it.
  if (arg(0) == 'user') {
    $variables['own_account'] = FALSE;
    // If we are on 'user' and are logged in then we are on our own user page.
    if (!arg(1) &amp;&amp; $variables['logged_in']) {
      $variables['own_account'] = TRUE;
    }
    // If we are on 'user/UID' then check against the current user.
    else if (is_numeric(arg(1)) &amp;&amp; !arg(2)) {
      global $user;
      if ($user-&gt;uid == arg(1)) {
        $variables['own_account'] = TRUE;
      }
    }
  }
}


Then you can use the variable $own_profile in your page-user.tpl.php template, which in this case might look something like this:

&lt;?php if ($own_account): ?&gt;
  &lt;div class="own-account-msg"&gt;
    Here is some markup for the message a user sees when they are on their own account page.
  &lt;/div&gt;
&lt;?php endif; ?&gt;