It depends on what your goal is. Different communication needs different practice styles.

I'm an American who emigrated some years ago to Eastern Europe and work in IT with a lot of persons from Eastern Europe, and we do some trainings on working with clients (particularly Americans and UK folk) in IT settings.

All my colleagues are near native fluent in English but what they sometimes struggle with is figuring out the nuances of effective business and technical communication, or both at the same time. Putting aside how a lot of technology terms get absorbed into non-English languages, the main issue is that the speech and stylizing of ideas in their native languages don't translate well to English.

The good news is that even for native English speakers, effective speaking is typically at a fairly low level, and business and technical talk have similar things to practice.

1. Ensure you're on top of articles in English and when to use which one. Even native speakers confuse them for otherwise educated persons and for non-native speakers, it tends to be noticed a bit more. (My Russian speaking colleagues struggle with this sometimes as they don't have articles really)

2. Focus on clarity of thought instead of specific high-level words. I personally think that tests like the TOEFL set non-english speakers up for disappointment as a lot of the items they check on are not common terms in every day speech.

3. English is a bit more "fluffy" than other languages which can come off as direct to native English speakers, and to make it worse, many English classes teach overly formal English to compensate for this. Go to a bar or out to eat with native speakers with the understanding that you want to practice with them and just study how they say things. Keep in mind many native English speakers don't remember the proper terms for rules of English grammar, and you may know it even more than they do. Instead, focus on just mimicking their style and asking how phrases sound to their ear (do keep in mind that everyone has a style of speech, so try to get a few sources to practice with)

4. I actually would advise against watching TV/Movies to learn how to improve communication -- these are highly stylized conversations with a pacing that is only found in TV/movies, you really will find it difficult to convert it. Some podcasts/youtubes might be better, but focus on the more casual channels as popular ones have a specific style meant to be attention grabbing to viewers. An example of one I think isn't bad is LockPicking Lawyer; he speaks about fairly technical things and definitely has settled on a calm and cadenced style, but at the same time the word choice and way of presenting is fairly normal and well received. Find ones like that.

5. Effective business communication is usually about brevity and presenting a large volume of information in a simple and evocative way; that is, you can say something like "our current estimates for the project have exceeded the initial projections and we expect to have to adjust the desired deadline to a more achievable goal". Or you can just say "The current evidence suggests we will miss our initially projected deadline, so we will need to adjust it to a more realistic point." The difference is adding in a lot of excess terminology that really isn't used in every day speech; the concepts are taught in business classes to allow for a teaching framework, but just like latin phrases aren't typically used in US legal proceedings, no one communicating effectively actually uses such terms unironically.

6. Follow the idea of Richard Feynman and first ensure you understand the concept of what you want to discuss. Feynman famously tried to explain a complex subject to first years and gave up saying: "I couldn't reduce it to the freshman level. That means we really don't understand it." This isn't specific to English, it's just general communication skills, so make sure you know how to explain to yourself what you want to say, and then you can find the words to further simplify difficult concepts to others.

7. This one is more a personal thing for me, but I advise be the one to ask your speaking partners "am I saying this right?" I've had many colleagues and friends tell me "please just outright correct me when I make a mistake in English", but it's pretty difficult to do in a "right" way, and it's to easy to have a misunderstanding that results in hurt feelings. If you're not sure on a phrase or term, just ask about it and your partners will be much more open to explaining.

8. Since you asked about conciseness, take time to write out your thoughts on complex subjects. Don't edit at this stage, just write. Once you're done, go back over it and start to strike out every extra word and sentence that doesn't contribute to the total understanding of your thoughts. Practice this a bunch and then start recording yourself on new subjects. If you have an opportunity for public speaking with your work, do it. It's scary at first, but you'd be amazed how fast someone can fall in love with it; I'm 100% an introvert, but I've come to love doing presentations on technical topics and people seem to really enjoy them. Planned speaking is really good because you typically have a time limit, and you have to first write what you want to say, see how long it takes you in practice, then cut it down. It's a great exercise that helps you to understand what can and cannot be removed to keep effective communication while keeping brief. Even just for an internal video series or small presentation to friends, it's a great exercise.

Hope at least some of this sparks some inspiration.