I guess I wasn't sure how they would install tack strips with the tiles on the floor?

drill the tackless and use aluminum anchors of they can not nail it.

Drill and Aluminum Anchors

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You don't want to be drilling through asbestos tiles. That falls in the category of damaging, pulverizing, creating dust, etc. There are very good adhesives you can use to attach the tack strips. They come in tubes, like caulk, and are available at any good carpet store. Alternatively, you could use those carpet squares that have pad attached. No tack strips required.

Have them try to nail it first.

Rare is the occasion you cant nail concrete strip through two layers of tile.

Dont glue the tackless. It will be a futile effort because the pressure of the stretch will most likely pull up the tile the tack strips are glued to.

Pre-padded carpet is certainly an alternative like Alice suggests.

Other options are shooting in a layer of plywood over the tile that you can nail tackless to and install carpet on if nothing else works.

We have successfully drilled through asbestos tile with slow rotary masonry drills/bits. This did not create dust from the tile because of its resiliency. It just flaked off chips that were easily shop vacced up. I do not recommend a high speed drill though. If the tile has resiliency then it may work for you, but if brittle, then may cause dust because of its hardness and is not recommended.

The dust from drilling is pretty much contained. I have successfully nailed through into concrete.
Glue would be futle.

The alkali in concrete, especially below-grade concrete, will destroy aluminum. Plastic anchor inserts would be a better choice.

Aluminum drives have been an industry standard for drilling tackstrip into concrete for many many years, and without issue I might add.

If excessive pH was in a slab it would not affect aluminum drives for the life cycle of any carpet.

Furthermore, plastic sleeve inserts can create humps under carpet at the tack strip because they can not be pounded flush (like aluminum drives can) without smashing the tackness, so you are left with tiny humps that may telegraph throgh some carpets, or tackstrip that ight break and pull apart during a stretch.