1. This differs more from workplace to workplace than industry to industry. Many dev workplaces will give developers a week or two of official "training time." Many other industries have workplaces that don't do that.
2. The reality is, training on new technologies has more value to you than it does to your employer. If your employer uses J2EE, Subversion, and Oracle, there's little benefit to them sending you off to learn Node, git, and Mongo. But there's a big benefit to you in knowing that, potentially. If the benefits of learning flow more to you than to your employer, you shouldn't be surprised if the burden of learning falls onto you.
3. But mostly, if you're a person who finds software development genuinely interesting (the hobbyist-type), you'll keep up on stuff basically recreationally. When you read about the latest whatever, you'll be curious and want to try it out and learn how it works because you're interested in that. You don't need any formal training to learn new technologies, just a little motivation and time. An hour spent not watching TV can get you a long way in understanding a new technology.