Free animated screensavers are generally safe. However, the overall safety largely depends on the screensaver's author and the program's source. It is recommended to rely on trusted websites before downloading and executing a screensaver.
Free animated screensavers may come with promotional offers and advertisements built-in, which are shown while the screensaver is active or at any given time after its installation. Read the screensaver's terms of service or its general instructions prior to downloading it, as its author may disclose his intentions in exchange for providing it free of charge. You may also search for the screensaver in question on various trustworthy websites before downloading. These include Cnet, American Greetings and other venues with an accessible and reliable history.