I was given to believe that he is missing since 18th June 1986

I am a American speaker. I don't think either sentence is good. The phrase "given to believe" can mean "inclined to believe", I am not even sure that that is correct.

That being said, the second sentence is more correct.

But I am not even sure what you are trying to say. Were you misled in some way? Were you led to believe that he was missing? Was this just a intuition?

The sentence isn't very clear.

In British English, "I was given to believe something" means "I was induced to believe something" or "I was told something", often in circumstances where we relied upon that information and later found it was wrong, e.g. I was given to believe there would be a car waiting for me at the airport, but there wasn't one, so I had to find a cab. From Google searches, I strongly suspect that this usage is mostly found in British English zones of the world. Whether we make this substitution of "was told" or not, the second sentence is the only correct one. We don't say that someone "is missing" since a date or time in the past, we say that he or she has been missing since that time.

It's commonly used in the USA as well.

I was led to believe...

To me that phrase would not mean "induced". It would mean to "have the tendency to".

"He was given to drink in the afternoon".

Yes, but that's different. We use 'given' that way in BrE also. Given to cursing in his sleep. But if I was given to believe, or given to understand something, that means I was told something.

Also, I gave him to understand, he gave me to believe, etc.

The situation is not as gloomy as we have been given to believe.

He had been summoned from Rome, and as soon as he docked at Portland, Maine, in early February, he had a request from Cartier to come to Ottawa for discussions. Taché received a copy of the December proclamation of amnesty, which he was given to believe covered every action that had taken place or might take.

Oxford Dictionary:


MacMillan Dictionary:

(This is given as both British and American usage)