I am talking about watching those shows in websites that provide them for free. I could of course afford to subscribe to paid websites such as Crunchyroll and the likes, but often I couldn't find the series I wanted to watch there. The free ones usually host a load of anime and drama titles.
Is watching free anime and Japanese drama in Japan considered safe?
In Japan or not, sites other than official sites that provides streaming for copyrighted/law protected contents are illegal and against the law. Thus, the answer to your question is no, it is not safe, it is against the law, it is a crime.
From IT (information tech) point of view, many such sites are actually crap sites that is full of ads, click baits, and links that will only redirect you around without showing the content if any. Such sites might also send adwares, malwares and the like to your computer. It might also be used to stole your identity. Thus it is also not safe from IT point of view.
If you are in Japan, however, you can watch the anime when it was broadcasted. This is perfectly legal. In fact, you can also record it if you want to watch later. AFAIK from anime that I watched, this is also legal. Some anime shows that recording a broadcasted show is okay. For example, in Futsuu no Joshikousei ga [Locodol] Yattemita., Yukari recorded a show that is being broadcasted. In Doraemon, Doraemon and Nobita would often record a show since it is being broadcasted at the same time as a baseball match that the father likes to watch.
Of course since it is limited to broad-casted anime only, you won't be able to watch older animes.
TL;DR: Yes. It is safe but with a catch.
Streaming services that have a license to broadcast content within Japan such as Niconico are ideal.
Streaming services meant to distribute anime abroad such as Crunchyroll and Hulu are also legal to view in Japan. That said, both sites have region filters because they are not licensed to show many series within Japan. When I visited Japan in April, 2016 the Crunchyroll's massive catalog was reduced to less than 20 items that were licensed for viewing.
Other unlicensed methods are both illegal and dangerous. The Japanese authorities take piracy very seriously. Illegal viewing could land you in prison for two years, even as a tourist. This includes content posted on sites without a license to broadcast that content in Japan, YouTube, downloads/file sharing, screen sharing (RabbitMQ), and using a VPN to get around region locking on sites with a license to broadcast elsewhere.
Unless the site you wish to watch on is sponsored and endorsed by the producers themselves, you can assume they are not safe. Though if you wish to do a little background check on the website, for example searching "is GenericAnimeStreamingSite.com legal" will do the job (in most cases).
No company wants their content, which could earn them money (like anime is intended to do) to be pirated. If you're watching on a third-party site which doesn't give money to the producers (or however they transfer funds to gain rights to anime), that is exactly what you're doing.
If you wish to be safe, the best thing to do is to subscribe to an official site, like Crunchyroll, as you said. Of course (as Memor-X reminded me), Crunchyroll as well as some others do offer free options, though those aren't optimal due to the ads. If you wish for the most hassle free and legal viewing experience, your best bet is simply to subscribe to a site like Crunchyroll. If affording it is no problem, then that's one of your options.
The other of course, would be to buy the Blu-Ray version, though that's more expensive.
