Just tested on the Note 3 as follows :-
When both sim 1 and sim 2 show 4G signal. Under normal SMS both sim 1 and sim 2 can send out the normal SMS but upon sending the phone will prompt the normal SMS to be send out by sim 1 or sim 2 number.
Upon either select sim 1 or sim 2 as "Internet" setting in the SIM cards & mobile networks. You can connect to the internet without problem and the sim 1 and sim 2 still show as 4G+4G,also sending WhatsApp and LINE without problem.
For MIUI 8.0.1.0 global version, under the SIM cards & mobile networks -> SIM CARD SETTING ,when select sim1 under MOBILE NETWORK, you have three choices to select - Access Point Names or Preferred network type or Mobile networks and in the Preferred network type, you can select Prefer LTE or Prefer 3G or 2G only but when you select sim2 under MOBILE NETWORK only has one selection - Mobile network.
So really cannot check the full details on each SIM status unless the global beta version 8.6.9.9 include the full checking display of both SIM card status.