System parameters in identity-based encryption In IBE schemes, the system parameters are $(q, \mathbb{G}, F, \hat{e}, P, Q, T, H_1)$. I don't know $\hat{e}$.

For example, in type A pairing…

type = a

q = 98826429041171753291515535532523512299028170537954154869719707264887274916552228805607584116490046284509883309001532457986879277885241872021906840932513241346999389365188296460009947
h = 32243626948934860887488490158437299489453513352745889246437755713701521031193083418924110592954582395114812811896992400310730276
r = 3064991081731777546575510593831386635550174528483098623

exp2 = 181
exp1 = 127

sign1 = -1
sign0 = -1


Are those system parameters? Does the server share these properties with clients?
The pairing description (type, q, h, r, etc.) is the definition of the field that the elliptic curve of some type operates on. The actual curve is identified by its type and is baked into the framework (e.g. PBC) you're using. 

This definition corresponds to $q$ (group order, same as q), $\mathbb{G}, \mathbb{G}_T$ (groups defined by pairing description and curve) and $\hat{e}$ (pairing function $\hat{e}:\mathbb{G}\times\mathbb{G}\rightarrow\mathbb{G}_T$). $\hat{e}$ greatly depends on the actual elliptic curve, because different algorithms or parameters are used. All the other parameters are unique to the scheme that is devised using bilinear pairing.

The pairing description is necessary to have for doing any calculation with the described scheme. It must be known to all participants. Of course, if you're always using the same pairing description, you don't have to send it along with every message of your system. It can be baked into the client software. 

Remember to generate a fresh and valid pairing description before you deploy your system into production. This makes it a little harder for attackers to break a system. If every one uses the same defaults, then pre-computation on that single group can break a lot of systems.