We have setup an Open Data site with Local Projections functionality enabled: Home | ArcČR® 500.
The data is published in local coordinate stystem (EPSG:5514) as map service on ArcGIS for Server: OpenData/AdministrativniCleneni_v12 (MapServer).
There is the corresponding datum transformation (WKID:1623) specified on the map service (MXD) to transform data between EPSG:5514 and EPSG:4326.
Downloading data as shapefile from the open data site, the resulting shapefile is in local projection (EPSG:5514), however, it is shifted from the original data (cca 120 meters). This shift is usually due to not using the correct datum transformation between EPSG:5514 and EPSG:4326 which is WKID:1623.
Observing the ArcGIS for Server log files, it seems that open data site requests data from the map service in EPSG:4326, then it transforms the data into EPSG:5514 not using the correct datum transformation and sends back the resulting shapefile.
I thought that Open Data's Local Projections functionality means that open data site requests data from the source in the source's coordinate system, while it seems that open data site provides the data projections on its own. If that's true, it should use the correct datum transformations.
I went to your link above and downloaded the lyr file and had a look in ArcGIS (v10.3.1).
Loaded the lyr file first into a new map doc, then loaded the topo basemap (WGS84 based).
I note that I did not get the usual warning about different datums and the need to apply a transformation.
Although when I look at the properties, it does indeed know that it is JTSK based. So, something is not quite getting through here.
I know that the online services are supposed to handle the transformations correctly, I do not think that esri have got this quite right yet.
In our experience, we have created WGS84 based data from the local datum based data, then published that to avoid potential confusion.
Might not be the answer you want to hear but you may have to reconsider your workflow.
The shift is obvious from this snapshot. No transformation applied.
The workflow you describe does not use transformation on the services level, but ArcMap is transforming the topographic basemap (map tiles) from Web Mercator (WGS84 based) into S-JTSK on its own. If you tell ArcMap to use the correct tranformation, the data is projected correctly.
To test the transformation on the service level, just open the map service in ArcGIS Map Viewer, instead of ArcMap. This way, the client (browser) requests data in Web Mercator and the map service projects data using the correct datum transformation which is set in the MXD used to create the map service.
If the open data site does not project data on its own but use the projection on the service level, datum transformation settings is in the hands of data provider.
Anyway, thank you for your answer.