Vtyjkyuk

The war induced provincial governments to establish their own system of free public employment offices.

I am trying to say that at the time of the war, provinces each established a system of employment offices. These offices aided in the distribution of labour for the war effort. The provinces did this on their own, they felt there was a need for such a system. They were obviously not the same, but had one key element that was common: they were labour exchanges that distributed man power.

The way I have written it sounds weird to me, esp. the part "their own". It kind of feels that using "their own" somehow conveys that the federal govt. beforehand set up these offices FOR the provinces. I will explain their purpose in following that sentence. Can someone help me figure out a way to restructure this please?

Reading the question, an answer, and all the comments, it's still not clear to me if you're actually trying to describe a set of individual systems or a single system.

Depending on what you're actually trying to describe, I would suggest one of the following:

1. The war induced each provincial government to establish its own system of free public employment offices.

Here, adding each and changing their own to its own clarifies that it's one system per province.

2. The war induced provincial governments to establish a joint system of free public employment offices.

Here, it's clear that it's only a single system—but one that was established through the joint effort of all provinces.

Using "their own" is correct since the provincial governments came up with the systems themselves. You could preface your sentence by saying

Without a formal labour exchange in place, the war induced provincial governments to establish their own system of free public employment offices.

This would remove the ambiguity you point out of whether the government already had something in place for the provinces.

The war induced provincial governments to establish their own system of free public employment offices.

The way it's currently worded, it sounds like all the provincial governments had the same system, since English, unlike for example French, tends to use the plural when multiple entities own one of the same thing each. Multiple things altogether, plural noun. (In French, one thing each, singular noun.)

The war induced provincial governments to establish their own systems of free public employment offices.

Using the plural makes it less likely (but still possible) that the provences were sharing systems. Words you can add to try and mitigate this effect are each and independent:

The war induced each provincial government to establish their own independent systems of free public employment offices.

The war induced each provincial government to establish their own systems of free public employment offices, independently of one another.

The war induced the provincial governments to establish their own systems of free public employment offices, independently of each another.