The cloth markets in Lüneburg in 1402 were governed by statutes including the requirement for each shop to provide a guard.

"Item eyn jewelik wantsnyder schal hebben enen wachter myt enem spete und mit enem ysern hode und mit ener luchten dar eyn licht inne barne vor siner wantkisten so lange bet de erste misse ut ys unde dat me van dage sen moge. Item ok schal eyn jewelik wantsnyder sinen wachter anrichten, dat he nynerlye unstur dede in den wanttelden und myt synen medewachteren."

"Each cloth cutter shall have a guard with a spear and a iron hat and a lantern with a burning candle in it, for his cloth chest as long as the first mass is done and the break of day can be seen. Also each cloth cutter shall instruct his guard, that he shall in no way behave uncouth in the cloth rooms and with his fellow guards."

Thanks to many visitors in the 19th century we have a pretty clear understanding of the Grand Bazaar in Istanbul, and armed guards were indeed mentioned. The Empire was segregated (in a way) on ethno-religious grounds who maintained a sort of quasi-independence in domestic trade affairs. As individual types of merchandise items were also segregated to their own streets and corners you had ethnicities dominating unique trades, with Jews, Armenians, and Greeks being prominent. Guilds formalized these trades and they did hire their own security to guard their own small sections of the market (though if the guilds cooperated with one another on shared security I couldn't find). I can't tell you how well they were received but I could only imagine it was fairly positive as theft in the bazaar was reportedly an exceptionally rare occurrence.

Gülersoy, Çelik, Story of the Grand Bazaar, 1980. 49-50.

Merchants in the princely era between Kievan Rus' and Novgorod in the many principalities had pretty interesting security. In Novgorod at least, most goods were displayed in the open or in warehouses, and couldn't be easily stolen or moved. Someone might steal a fresh fish for example, but stealing an entire fur during trading hours? Not going to happen easily.

There were however private security for important merchants, the Kynaz, Boyars et all; think of them as the kind of private corporate security you might see in some really well-to-do outlets. Similarly, a lot of the warehouses that housed goods would have full-time security details, either hired (and usually 'foreign', even if that just meant a subgroup that wasn't East-Slavic) or recruited from local cohorts called Druzhina. Pay could be very lucrative and the job somewhat prestigious during the glory days, but dwindled pretty rapidly as the age of Principalities faded into the early modern era. For example, as the grain trade was cut off from Novgorod, guards who had been loyal and well-paid would end up stealing from storehouses simply to make ends meet, damaging morale all around.

Short(ish) version is, during days of wealth, they could be all but invisible and the markets would be largely open-air and unguarded save for local military districts. During days of poverty or war, guards might be stationed more regularly, but also suffered pretty heavily from the dwindling fortunes of the private clients - largely mercantile interests - that employed them.

For 19th century France Emile Zola's book Le Ventre de Paris might be an interesting source. The book plays around the massive Parisian markets called Les Halles and the writer broadly describes the massive size of it. In the book it is mentioned how there is a double control, the prefecture of the Seine and the Prefecture of Police. The latter licences the salesmen and maintains the right of supervision over them while the local prefecture has an interest in transactions since they are taxable. What is striking is how in the rest of the book one hears little about anyone other than the prefect who is controlling the market. The main character who got this role spends a lot of his time keeping the peace in the market halls. Burglary isn't mentioned a lot but what is mentioned is the power of the stand-holders. This would lead one to believe that most burglary would have been policed by the stand-holders themselves.