I definitely pronounce the "th," and I think most native speakers do, too. It may be a little softer than when the word "both" is pronounced in other contexts.
Here is a recording of myself:
"(1) I've looked at clouds from both sides now. (2) To those nations who would make themselves our adversaries, we offer not a pledge but a request: that both sides begin anew their quest for peace. (3) Lay the fish on an oiled gridiron, and broil it on both sides."
Here, to help you hear the difference, I've recorded it two ways, using the phrase "broil it on both sides." First, I say it as I normally do; then I leave out the "th" and pronounce it as "broil it on bow sides." I repeat this three times.
If you are having trouble pronouncing it, and want to know whether it is acceptable to just leave out the "th" and say "bow sides," yes, you will be understood and in informal conversations native speakers may not even notice it. At worst they will say "his diction isn't perfect."
For a illustration of other people saying "both sides," consider one of my favorite songs, "Both Sides, Now," which you can find on YouTube sung by a variety of singers. For example, listen to Pete Seeger, the great singer of the 1960s folk revival:
The most famous version of it is probably by Judy Collins. Here she's singing it live. The "th" is very definite to my ear:
Here, it's sung by Joni Mitchell, who wrote the song, and again, I've chosen a live version. The "th" sound is less obvious than with Pete Seeger and Judy Collins, but it's there.