I wonder, how wise is it to allow Chrome and Firefox to a) remember the passwords b) synchronize them? My gut tells me that if it's not man in the middle who can intercept them, but Google and Mozilla themselves can see them on their servers or with help of their browsers. Of course, they say they won't and the passwords are stored encrypted, but can we know that for sure? Maybe the browsers themselves secretly send the passwords to Google and Mozilla. 

I've just begun using keepass recently, therefore at least I have a place where my passwords are stored locally, because previously I  stored them only in the  browsers and synchronized. And now I think I shouldn't  synchronize them anymore.

To expand on what @d1str0 said: if the creator of your browser wanted to steal your passwords, it would be trivial to send them to a manufacturer controlled server whenever you entered them - they don't need to bother with the hassle of telling you about sync procedures, or offering to remember passwords. All browsers by default send a certain level of usage data back, usually crash reports and update checks, which could easily conceal password and username data.

However, if any browser was found to be doing this, there would be outcry against that manufacturer - look at the rage directed at Microsoft following the release of Windows 10 with the reporting back enabled there.

Keepass and Password Safe are both open source (so, given sufficient programming knowledge, and a trusted compiler, you can be sure they're doing what they say they are, and nothing else - sufficient programming knowledge may well be a very high level though). In both cases, the encrypted password files should be safe to sync, even to third party sources, as long as the safe password is not provided. Breaking AES (Keepass) or TwoFish (Password Safe) without the appropriate key (the safe password) comes down, as far as we know, to brute force.

Lastpass and 1Password both require you to trust the developers, and sync by default to a remote location. Theoretically, they are safe, but there wouldn't be any obvious way to detect a vulnerability in them relating to storage. If you're concerned about Chrome or Firefox stealing passwords, logically, the same arguments apply to these apps.

Personally, I use one of the cloud based password services mentioned - I've considered the risks and benefits, and balanced the amount of security I'm willing to accept against the ease of use for the service, and decided that for my use cases, it's acceptable. Your acceptable risk might well be different - if you consider AES as vulnerable, for example, keeping a Keepass safe on an encrypted USB key which uses a different encryption algorithm might be a viable option, but uploading the file to a third party service might be "too risky" for you.

It comes down to what you consider safe, having evaluated the options. Many security professionals have considered this problem though, and generally advise using password safe type software over allowing browsers to remember passwords, simply because browsers used to be terrible at it - they allowed access without a master password, and used poor encryption methods. Some of these issues have been addressed now, but old habits die hard!
