@PotomacBobI have played with both over the years.
But most commonly, as quite a few of my stories have people who are not native speakers of English, I simply show this in how they talk. Words transposed, articles of speech completely dropped, even sometimes using a word somewhat inappropriately. And having listened to many not from the US for decades, I think it adds a little variety to the story. And I quite often will never really say where they are from, or until much later. But how they talk and what they say should make it obvious.
"Oh no my friend. Propaganda, but not lies, truth. Propaganda, it not mean what you think. What is word, spin! Yes, spin. Radio Moscow not tell lies, but does use spin for favor them, just as US news does. Here is thing, US news is garbage. It only covers things they want US people to know. Here I know more, because I listen to Radio Moscow. Here is thing, I know something coming months ago."
I looked at him, and he nodded. "Iraq borrow a lot of money when they fight Iran, they owe $14 billion to Kuwait. Iraq in trouble, lots damaged in fighting Iran, say they can not repay money owed. They ask Kuwait to forgive debt as friends. Kuwait tell them to pound sand. Now if this like China, Iraq may just say they proving point, take some things, destroy some things, then go home and ask again to forgive debt. They prove point, no need to stay."
To anybody familiar with how somebody from Eastern Europe sounds when they talk, this should stand out as to where this character is from. He is clear, but a bit choppy and off in how he phrases things, but it is in keeping with his original language.
And also the same with "era speech". I have several set in past decades, and I try hard to capture that era in what they say and how they say it. In at least 2 of my them (one set in the 1920's) I even added a glossary to explain what some of those old phrases meant.
"Oh, we were forked out by the airport, and the damned scumbag broke. Neither of us knew until it was over, and saw he had put his mess inside of me instead of the lambskin."
"What, you two were making whoopie?" she said, a bit scandalized.
Of course, in more modern "American English" I would have said it differently, but that would have lost the flavor of the era. Now, "fucking" or "screwing" would have replaced "forked" and "making whoopie", and "rubber" or "condom" for "scumbag" and "lambskin".
I guess one advantage of living in LA for so long and traveling all over in the military (as well as being married to an immigrant for decades) is that I am familiar with how different people talk, and capturing what is said in my mind and into the story.