This question has been answered by BobRay. See the first response.
[[!Login? &loginResourceId=`53` &logoutResourceId=`54` &loginTpl=`lgnLoginTpl_cs`]]
No. Each browser maintains its own cookies. Do you have the snippets cached, or uncached? If you have them cached, then their output will be cached in the page, no matter what your login status is. Login and registration snippets must be called uncached - [[!Login? ... ]]
Setting up permissions can be difficult at first. I don't do it very often, so it still takes me a few tries to get it right when I do need to use them.
[[!Login]]
<div class="loginMessage">[[+errors]]</div>
<form action="[[~[[*id]]]]" method="post" class="form">
<label class="loginUsernameLabel">[[%login.username? &namespace=`login` &topic=`default` &language=`cs`]]
<input type="text" name="username" class="cfinput">
</label>
<label class="loginPasswordLabel">[[%login.password? &namespace=`login` &topic=`default` &language=`cs`]]
<input type="password" name="password" class="cfinput">
</label>
<input class="returnUrl" type="hidden" name="returnUrl" value="[[+request_uri]]">
[[+login.recaptcha_html]]
<input class="loginLoginValue" type="hidden" name="service" value="login">
<input type="submit" class="button eshop" name="Login" value="[[+actionMsg]]">
</form>
[[!Login? &loginResourceId=`53` &logoutResourceId=`54` &loginTpl=`lgnLoginTpl_cs`]] [[-testlogin]]
[[!Login]]