Well … first you need to only download the software from the official links of the provider, eg LastPass, Dashlane or 1Password, or from the app store of your mobile device.
Then be assured that the passwords (if you use these downloads) are not stored on a central server anywhere. They are stored in encrypted form on your device. When you open a different device the password is transferred from one “vault” to another in encrypted form. The service provider just provides the encryption algorithm which it can’t have access to itself. So rest assured as long as you use the “official” downloads they’re very safe.